Answer:
[tex]a. \;\; \boxed{\$ 530.60}[/tex]
[tex]b.\; \;\boxed{ \$552.45}[/tex]
[tex]c,\;\;\boxed{\$610.60}[/tex]
[tex]d. \;\; \boxed{\$745.91}[/tex]
Step-by-step explanation:
All four cases deal with compound interest on the same amount of $500 and same interest rate of 2%.
The only difference is in the frequency of compounding and time
Compound Interest Formula
[tex]\boxed{A = P\left(1 + \dfrac{r}{n}\right)^{n\cdot t}\\\\}[/tex]
where
In this particular problem we have
P = $500
r = 2% = 0.02
These are common for all parts of the question
Only n an t are different for each of the question sub-parts
Part a
Compounding is done annually (once a year) for 3 years
n = 1
t = 3 years
n · t = 3
[tex]A = 500\left(1 + \dfrac{0.02}{1}\right)^3\\\\A = 500\left(1.02\right)^3\\\\A = 500(1.061208)\\\\A=\boxed{\$ 530.60}[/tex]
For accuracy of calculations, I will not compute and store the exponent part, I will perform the calculations in one shot
Part b
Here the compounding is done quarterly (4 times a year) for 5 years
n = 4
t = 5 years
nt = 4 · 5 = 20
[tex]A = 500\left(1 + \dfrac{0.02}{4}\right)^{20}\\\\\\A = 500(1.005)^{20}\\\\A =\boxed{ \$552.45}[/tex]
Part c
Compounding done monthly(12 times a year) for 10 years
n = 12
t = 10
nt = 120
[tex]A = 500\left(1 + \dfrac{0.02}{12}\right)^{120}\\\\\\A = \boxed{\$610.60}[/tex]
Part d
First let's figure out what continuous compounding means
[tex]\fbox{\begin{minipage}[t]{1\columnwidth \fboxsep - 2\fboxrule}%\textsf{What is continuous compounding?} \\\textsf{Continuous compounding is the mathematicallimit that compound interest can reach if it's calculated and reinvestedinto an account's balance over a theoretically infinite number ofperiods. While this is not possible in practice, the concept of continuouslycompounded interest is important in finance. (Investopedia)}\}%\end{minipage}}[/tex]
The formula for continuous compounding can be determine by using the standard formula for periodic compounding and taking limits as
[tex]n \rightarrow \infty[/tex]
Therefore, for compounding continuously , the formula can be derived from
[tex]\lim _{n\to \infty } P\left(1 + \dfrac{r}{n}\right)^{nt}\\\\[/tex]
One of the limit formulas states
[tex]\lim _{x\to \infty } \left(1 + \dfrac{a}{x}\right)^{x} = e^a\\\\[/tex]
Therefore
[tex]\lim _{n\to \infty } \left(1 + \dfrac{r}{n}\right)^{n} = e^r\\\\[/tex]
So for the continuous compounding case, the formula is
[tex]\boxed{A = P \cdot e^{rt}}[/tex]
Here we have
r = 0.02
t = 20 years
rt = 20(0.02) = 0.4
Plugging in P = 500, and rt = 0.4 we get
[tex]A = 500 \cdot e^{0.4}\\\\A = \boxed{\$745.91}[/tex]
Answer:
a) $530.60
b) $552.45
c) $610.60
d) $745.91
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Compound Interest Formula}\\\\$ A=P\left(1+\frac{r}{n}\right)^{nt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $n =$ number of times interest is applied per year \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}[/tex]
Part aGiven:
P = $500r = 2% = 0.02t = 3 yearsn = 1 (annually)Substitute the given values into the compound interest formula and solve for A:
[tex]\implies A=500\left(1+\dfrac{0.02}{1}\right)^{1 \cdot 3}[/tex]
[tex]\implies A=500\left(1.02\right)^{3}[/tex]
[tex]\implies A=500(1.061208)[/tex]
[tex]\implies A=530.604[/tex]
Therefore, the value of the account after 3 years if interest is compounded annually is $530.60 (nearest cent).
Part bGiven:
P = $500r = 2% = 0.02t = 5 yearsn = 4 (quarterly)Substitute the given values into the compound interest formula and solve for A:
[tex]\implies A=500\left(1+\dfrac{0.02}{4}\right)^{4 \cdot 5}[/tex]
[tex]\implies A=500\left(1.005\right)^{20}[/tex]
[tex]\implies A=500(1.10489557...)[/tex]
[tex]\implies A=552.447788...[/tex]
Therefore, the value of the account after 5 years if interest is compounded quarterly is $552.45 (nearest cent).
Part cGiven:
P = $500r = 2% = 0.02t = 10 yearsn = 12 (monthly)Substitute the given values into the compound interest formula and solve for A:
[tex]\implies A=500\left(1+\dfrac{0.02}{12}\right)^{12 \cdot 10}[/tex]
[tex]\implies A=500\left(1.0016666...\right)^{120}[/tex]
[tex]\implies A=500(1.22119943...)[/tex]
[tex]\implies A=610.599716...[/tex]
Therefore, the value of the account after 10 years if interest is compounded monthly is $610.60 (nearest cent).
Part d[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Continuous Compounding Interest Formula}\\\\$ A=Pe^{rt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\\phantom{ww}$\bullet$ $P =$ principal amount \\\phantom{ww}$\bullet$ $e =$ Euler's number (constant) \\\phantom{ww}$\bullet$ $r =$ annual interest rate (in decimal form) \\\phantom{ww}$\bullet$ $t =$ time (in years) \\\end{minipage}}[/tex]
Given:
P = $500r = 2% = 0.02t = 20 yearsSubstitute the given values into the continuous compounding interest formula and solve for A:
[tex]\implies A=500e^{0.02 \cdot 20}[/tex]
[tex]\implies A=500e^{0.4}[/tex]
[tex]\implies A=500(1.49182469...)[/tex]
[tex]\implies A=745.912348...[/tex]
Therefore, the value of the account after 20 years if interest is compounded continuously is $745.91 (nearest cent).
athletes in a particular sport are classified as either offense or defense. the distribution of weights for the athletes classified as offense is approximately normal, centered at 200 pounds, and ranges from 150 pounds to 250 pounds. the distribution of weights for the athletes classified as defense is approximately normal, centered at 300 pounds, and ranges from 250 pounds to 350 pounds. there are 1,000 athletes in each classification. which of the following is the best description of a histogram of the weights of all 2,000 athletes?
A histogram of the weights of all 2,000 athletes would be a graph with two normal distributions, one for offense and one for defense.
The offense distribution would have a bell shaped curve centered at 200 pounds, with a range of 150-250 pounds. The defense distribution would have a bell shaped curve centered at 300 pounds, with a range of 250-350 pounds. Each of the distributions would have the same area, representing 1,000 athletes in each group. The total area of the histogram would represent the combined 2,000 athletes with weights ranging from 150-350 pounds. It is important to note that the histogram would not represent the exact weights of the athletes, but rather the relative frequency of each weight.
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Is this sequence arithmetic or geometric?
List at least 5 terms of the sequence
Write a recursive definition for the sequence
This formula states that each term in the sequence is equal to the previous term plus 3.
Is this sequence arithmetic or geometric?This sequence is arithmetic, because the difference between consecutive terms is consistent.The first five terms of the sequence are 1, 3, 5, 7, and 9.The recursive definition of the sequence is a(n) = a(n-1) + 2, where a(1) = 1.This sequence appears to be arithmetic, as the difference between consecutive terms is constant.This can be seen by looking at the graph, where the y-values of each point are regularly spaced apart.The first five terms of the sequence are 2, 5, 8, 11, and 14.A recursive definition of the sequence is a formula that can be used to generate subsequent terms of the sequence.In this case, the recursive definition of the sequence is a(n) = a(n-1) + 3, with a(1) = 2.For example, the fourth term in the sequence (11) is equal to the third term (8) plus 3, and the fifth term (14) is equal to the fourth term (11) plus 3.This sequence appears to be arithmetic, as each consecutive term increases by the same amount. The difference between each term is constant, so this is a defining characteristic of arithmetic sequences.The first five terms of this sequence are 0, 2, 4, 6, 8.\A recursive definition of this sequence is a formula that describes the relationship between each term and the preceding term. The recursive definition of this sequence is an_n = a_(n-1) + 2, where a_n is the nth term of the sequence and a_1 = 0. This definition states that each term is two more than the previous term.To learn more about the sequence refer to:
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Consider the slope field shown: (a) For the solution that satisfies y(0) 0,sketch the solution curve and estimate the following: y(I) ~ 0.75 and y(-1) 0.5 (b) For the solution that satisfies Y(O) = 4, sketch the solution curve and estimate the following: Y(0.5) -0.5 and Y(-1) 0.75 (c) For the solution that satisfies Y(O) = -1, sketch the solution curve and estimate the following: Y(I) ~ and y(-1) ~
For the solution that satisfies, sketch the solution curve and estimate the following y(0)=0: y(1)≈0.75, y(-1)≈0.5; Y(0)=4: Y(0.5)≈-0.5, Y(-1)≈0.75; Y(0)=-1: Y(1)≈-1.5, Y(-1)≈-1.5.
(a) To sketch the solution curve for the given slope field, beginning at the point (0, 0) and following the arrows, draw a curved line that gradually moves from left to right, gradually increasing in value and then slowly decreasing. The estimated value of y(1) is 0.75 and y(-1) is 0.5.
(b) To sketch the For the solution that satisfies, sketch the solution curve and estimate the following y(0)=0: y(1)≈0.75, y(-1)≈0.5; Y(0)=4: Y(0.5)≈-0.5, Y(-1)≈0.75; Y(0)=-1: Y(1)≈-1.5, Y(-1)≈-1.5. curve for the given slope field, beginning at the point (0, 4) and following the arrows, draw a curved line that gradually moves from right to left, gradually decreasing in value and then slowly increasing. The estimated value of Y(0.5) is -0.5 and Y(-1) is 0.75.
(c) To sketch the solution curve for the given slope field, beginning at the point (0, -1) and following the arrows, draw a curved line that gradually moves from left to right, gradually increasing in value and then slowly decreasing. The estimated value of Y(1) is and y(-1) is -1.5.
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someone please answer and explain how to do this in steps i will give you a cookie
17. The equation can be rewritten as 2x²-8x+1=0.
18. The axis of symmetry is x = 5. The parabola opens up and has a maximum at the vertex (5, 6). The quadratic equation has two solutions, x = 4 and x = 6.
How many solutions does the graph have?17.
The equation can be rewritten as 2x²-8x+1=0.
Vertex:
The vertex is (4, -1). A minimum.
Axis of symmetry:
The axis of symmetry is x = 4.
The parabola opens up.
The graph has 1 solution.
18. How many solutions does the quadratic
have?
Answer: The axis of symmetry is x = 5, which is calculated by taking the coefficient of x (which is -10) and dividing it by 2. This is because the axis of symmetry is always in the middle of the two x-intercepts. The parabola opens up, which means it has a maximum at the vertex. The vertex is (5, 6), which is calculated by substituting x = 5 into the equation and solving for y.The quadratic equation has two solutions, x = 4 and x = 6. To find the solutions, one must set the equation equal to 0 and solve for x. This can be done by either factoring or using the quadratic formula. After finding the x-intercepts, the solutions can be determined by plugging in the x-intercepts into the equation and solving for y.To learn more about Solutions in the graph refer :
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17. The equation can be rewritten as 2x²-8x+1=0. 18. The axis of symmetry is x = 5. The parabola opens up and has a maximum at the vertex (5, 6). The quadratic equation has two solutions, x = 4 and x = 6.
17. The equation can be rewritten as 2x²-8x+1=0.
Vertex:
The vertex is (4, -1). A minimum.
Axis of symmetry:
The axis of symmetry is x = 4.
The parabola opens up.
The graph has 1 solution.
18. The coefficient of x, which is -10, is multiplied by two to determine the axis of symmetry, which is x = 5.
This is due to the fact that the axis of symmetry is always located halfway between the two x-intercepts.
The parabola widens, indicating that the vertex is where its maximum is located. The vertex is (5, 6), which is determined by solving for y while changing x = 5 in the equation.
There are two answers to the quadratic equation: x = 4 and x = 6. The equation must be made equal to 0 in order to get the solutions.
Either factoring or applying the quadratic formula can be used to accomplish this.
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Marissa bought 7 ounces of yogurt for $3.50.
What is the unit price?
$0.35 per ounce
$0.50 per ounce
$3.50 per ounce
$5.00 per ounce
Answer:
$0.50
50 cents per ounce.
Step-by-step explanation:
To get the unit price, you take the cost of the package divided by the parts it has. Since Marissa has a package of yogurt that costs $3.50 and there are seven ounces in the package we have this equation:
($3.50)/(7 ounces)= $0.50 = 50 cents = the second answer available.
Show that the two-variable function f given by f(x, y) = 2x^2 − xy is
differentiable at any point (a, b). What is the derivative of f at (a, b)?
Step-by-step explanation:
To show that the function f(x,y) = 2x^2 − xy is differentiable at any point (a,b), we need to show that the partial derivatives of f with respect to x and y exist and are continuous at that point.
The partial derivative of f with respect to x is:
∂f/∂x = 4x - y
The partial derivative of f with respect to y is:
∂f/∂y = -x
We can see that both of these partial derivatives are continuous functions of x and y and are defined for all (x,y) in the domain of f. Therefore, f is differentiable at any point (a,b).
The derivative of f at (a,b) is the gradient vector of f evaluated at (a,b), which is given by:
Gradient vector of f(a,b) = ∇f(a,b) = ( ∂f/∂x, ∂f/∂y ) = (4a-b, -a)
So the derivative of f at (a, b) is (4a-b, -a).
question approximately 38 percent of people living in region w have the blood type o positive. a random sample of 100 people from region x revealed that 35 people in the sample had the blood type o positive. consider a hypothesis test to investigate whether the percent of people in region x with o positive blood is different from that of in region w. which of the following is the appropriate null hypothesis for the investigation? Ha: proportion 0.38 Ha: proportion >0.38 Ha: proportion =0.38 O Ha: proportion <0.38
Null hypothesis for the investigation Ha=0.38
The null hypothesis is a kind of hypothesis which explains the population parameter whose purpose is to test the validity of the given experimental data.
This hypothesis is either rejected or not rejected based on the viability of the given population or sample.
In other words, the null hypothesis is a hypothesis in which the sample observations results from the chance.
It is said to be a statement in which the surveyors wants to examine the data. It is denoted by H0.
Here, the hypothesis test formulas are given below for reference.
The formula for the null hypothesis is:
H0: p = p0
The formula for the alternative hypothesis is:
Ha = p >p0, < p0≠ p0
The formula for the test static is:
[tex]z=\frac{p-p0}{\sqrt{\frac{p0(1-p0)}{n} } }[/tex]
Remember that, p0 is the null hypothesis and p – hat is the sample proportion.
38 percent of people living in region w have the blood type o positive.
w region
38% - o positive
x region
100 random sample
35 o positive
means 35/ 100*100
35% 0 positive
a hypothesis test to investigate whether the percent of people in region x with o positive blood is different from that of in region w
The formula for the null hypothesis is:
H0: p = p0 =0.38
Ha = p >p0, < p0≠ p0
Ha=0.38
Null hypothesis for the investigation Ha=0.38
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Use the distributive property to write each expression as an equivalent
algebraic expression.
6(4+x)
Answer:
= -6x + 24
Step-by-step explanation:
6(4-x) = 6*4 + 6*-x
= 24 - 6x
= -6x + 24
g the space complexities of bfs and dfs are o(b^d) and o(bd) respectivaly, why is one exponential with respect to d and the other not
BFS has an exponential space complexity with respect to d because it stores all the nodes at the same depth level before moving on to the next level
DFS doesn't have an exponential space complexity with respect to d because it only stores the nodes that need to be visited next.
The space complexity of breadth-first search (BFS) is O([tex]b^d[/tex]) and the space complexity of depth-first search (DFS) is O(bd), where b is the maximum branching factor of the search tree and d is the maximum depth of the tree.
The reason for the difference in the space complexities is that BFS uses a queue to store the nodes to be visited, while DFS uses a stack.
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Rework problem 27 from section 1.2 of your textbook, about the three subsets X1, X2, and X3 that partition a set X, except assume that the number of elements in X, is 4 times the number of elements in X2, the number of elements in X3 is 6 times the number of elements in X2, and n(X) = 99. (1) n(X1) = (2) n(X2) = (3) n(X3) =
The value of three subsets is [tex]n(x_{1})=36, n(x_{2})=9, & n\left(x_3\right)=54[/tex].
A set is a collection of objects or elements, grouped in the curly braces, such as {a, b, c, d}. If a set A is a collection of even number and set B consists of {2,4,6}, then B is said to be a subset of A, denoted by B⊆A and A is the superset of B.
The elements of sets could be anything such as a group of real numbers, variables, constants, whole numbers, etc. It consists of a null set as well.
If set A has {X, Y} and set B has {X, Y, Z}, then A is the subset of B because elements of A are also present in set B.
Let the number of elements in [tex]$x_2$[/tex] be [tex]$x$[/tex], then
[tex]$$\begin{aligned}& n\left(x_1\right)=4 x \\& n\left(x_2\right)=x \\& n\left(x_3\right)=6 x\end{aligned}$$[/tex]
Given that number of elements in [tex]$x=99$[/tex]
[tex]$$\begin{aligned}\Rightarrow & n\left(x_1\right)+n\left(x_2\right)+n\left(x_3\right)=99 \\\Rightarrow & 4 x+x+6 x=99 \\\Rightarrow & 11 x=99 \\\Rightarrow & x=9 \\\therefore & n\left(x_1\right)=4 x=4 \times 9=36 \\& n\left(x_2\right)=x=9 \\& n\left(x_3\right)=6 x=6 \times 9=54\end{aligned}$$[/tex]
Therefore, the value of [tex]& n\left(x_3\right)[/tex][tex]=54[/tex].
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Which graph shows the new image of the rectangle after a translation of two units to the left?
The new image of the rectangle after a translation of two units to the left is of option D
How to draw a shifted figure if the shifts are provided?Suppose the graph is drawn on the coordinate plane.
Let the shifting be (x,y) → (x+a, y+b)
Then, add 'a' to all x coordinates of the graph's points. Add 'b' to all y coordinates of the graph's points.
The resultant set of new points will be plotted.
Given;
A rectangle area on graph with coordinates (1,0), (6,0),(6,2), (1,2)
Now after the shift of 2 unit left, the x coordinates will be affected;
=(1-2,0), (6-2,0),(6-2,2), (1-2,2)
=(-1,0), (4,0),(4,2), (-1,2)
Therefore, the coordinates of shifted graph will be (-1,0), (4,0),(4,2), (-1,2)
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sam buys a computer for $590.00 plus $32.45 in sales tax find the sales tax rate?
Answer:
Is this what youre asking?
Step-by-step explanation:
Price before tax: $ 590.00
+ Sales tax (32.45%): $ 191.46
Total price including tax: $ 781.46
Summarize what these bond funds have in common.
Vanguard and Fidelity
There are many similarities between Vanguard and Fidelity, but there are also some significant distinctions. Vanguard focuses primarily on long-term, buy-and-hold investing.
What are bond funds?Fidelity, on the other hand, caters to investors who prefer a more hands-on approach.
There are more than 8,300 domestic investment-grade bonds held by the Vanguard Total Bond Market ETF.
Therefore, You'll need to study two (or more) sets of statements, keep track of several phone numbers, access a number of websites, and comprehend and keep track of hundreds of different funds.
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There are many similarities between Vanguard and Fidelity, but there are also some significant distinctions. Vanguard focuses primarily on long-term, buy-and-hold investing. Fidelity, on the other hand, caters to investors who prefer a more hands-on approach.
What do you mean by Bond?Bonds are issued by borrowers to attract capital from investors ready to extend a loan to them for a specific period of time. When you purchase a bond, you are making a loan to the issuer, which could be a corporate, government, or municipality.
By purchasing a bond, you are effectively lending the issuer money. In exchange, they commit to repay you the face amount of the loan on a particular date and to make periodic interest payments—typically twice a year—along the way. Unlike stocks, which grant you ownership rights, bonds issued by corporations do not.
Therefore, Vanguard's primary investment strategy is long-term buy-and-hold investing. On the other hand, Fidelity caters to investors who favor a more active strategy.
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Which equation has a solution of x= 3/4
The equation has a solution of x= 3/4 will be B. 8x = 6
How to calculate the equation?An equation simply has to do with the statement that illustrates the variables given. In this case, it is vital to note that two or more components are considered in order to be able to describe the scenario.
In this case, the value of x in 3x = 6 will be:
x = 6 / 3
x = 2
The value of x in 8x = 6 will be:
x = 6 / 8
x = 3 / 4
The value of x in 4x = 2 will be:
x = 2/4
x = 1/2
The value of x in 3x = 15 will be:
x = 15 / 3
x = 5
The correct option is B.
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Complete question
Which equation has a solution of x= 3/4?
A. 3x = 6
B. 8x = 6
C. 4x = 2
D. 3x = 15
For each of the following, count the number of four-digit numberssatisfying the condition and with digits in {1, 2,3, 4,5, 6}:(a) The digits are distinct(b) The number is even(c) The number is even and the digits are distinct.
(a) The number of four-digit numbers with distinct digits in {1, 2, 3, 4, 5, 6} is 6543 = 360.
(b) The number of four-digit numbers in {1, 2, 3, 4, 5, 6} that are even is (3543)/2 = 270.
(c) The number of four-digit numbers in {1, 2, 3, 4, 5, 6} that are even and have distinct digits is (321*1)/2 = 3.
Four-Digit Numbers CountingFor (a) the total number of four-digit numbers with digits in {1, 2, 3, 4, 5, 6} is 6 options for the first digit, 5 for the second, 4 for the third and 3 for the last one. So, 654*3 = 360.
For (b) since digits in {1, 2, 3, 4, 5, 6} the even digits are 2,4,6. for each of the even digits we have 5 options for the second digit, 4 for the third and 3 for the last one. In total, we have 354*3= 270 even numbers.
For (c) The total number of four-digit numbers with distinct digits that are even is 211 = 2, since there are 2 even digits in {1,2,3,4,5,6} and only one of each can be used. However, this count is double-counting the numbers, so we divide by 2 to get 3.
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In the figure the boundary of the shaded region consists of four semicircular arcs, the smallest two being equal. If the diameter of the largest is 14 cm and of the smallest is 3.5 cm, find1 . The length of the boundary2 . The area of the shaded region
The area of the shaded region and the length of the boundary is 86.59 cm^2 and 44cm respectively.
A semicircle is a shape that is half of a circle. It is defined by a center point and a radius, and it is made up of an arc that is 180 degrees of the full circle.
The circumference of a semicircle is given by the formula: C = π * d, where C is the circumference and d is the diameter of the semicircle.
Diameter of the biggest semi-circle = 14cm.
Diameter to two small semi-circles = 3.5cm.
Diameter of other semi-circle = 14 - 2(Diameter to two small semi-circles)
=14 - 7 cm
= 7cm
Length of boundary = Circumference of bigger semi-circle + Circumference of small semi-circle + 2 ×circumference of the smaller semi-circles
∴Length of boundary= [tex]\pi R+\pi r_{1}+2\pi r_{2}[/tex]
=> Length of boundary=[tex]\pi (R+r_{1})+2\pi r_{2}[/tex]
=> [tex]\frac{22}{7}(7+3.5)+2X\frac{22}{7}X1.75[/tex]
=> 44cm
Area of the biggest semi-circle = [tex]\frac{1}{2}\pi (7)^{2} cm^{2}[/tex]
Area of two small semi-circles = [tex]\pi (1.75)^{2}cm^{2}[/tex]
Area of other semi-circle= [tex]\frac{1}{2}\pi (3.5)^{2}cm^{2}[/tex]
Area of shaded region= π×(24.5−3.0625+12.25)
=π×27.5625 cm^2 =86.59 cm^2
Therefore, the area of the shaded region and the length of the boundary is 86.59 cm^2 and 44cm respectively.
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On the coordinate plane, the segment from D(-6, 3) to E(2, 3) forms one side of ADEF. The
triangle has an area of 28 square units. Select all of the points where F could be.
(5,-4)
(1, 10)
(7,4)
(-6, 10)
On the coordinate plane, the segment from D(-6, 3) to E(2, 3) forms one side of ADEF. The possible point F are (5,-4), (1, 10), (7,4), (-6, 10).
How many points does a F earn?Since we are aware that the triangle's area is 28 square units and that the base is the segment DE, which has a length of 8 units, we can construct the equation:
Area equals (1/2) × base × height = (1/2) × 8 × height=28
We can calculate the height and discover that it is 7 units.
Point F must be situated 7 units above or below the line DE since the triangle's height is perpendicular to the base DE. Therefore, any position that is 7 units above or below the line DE, which is determined by the equation y = 3, is considered to be at point F.
So the possible point F are (5,-4), (1, 10), (7,4), (-6, 10)
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Complete question -
On the coordinate plane, the segment from D(-6, 3) to E(2, 3) forms one side of ADEF. The
triangle has an area of 28 square units. Select all of the points where F could be. (5,-4), (1, 10), (7,4),(-6, 10)
There exist two complex numbers c, say c1 and c2, so that 2+2i, 5+i, and c form the vertices of an equilateral triangle. Find the product c1c2 .enter image description here
So far, I have used the distance formula to get (a−2)2+(b−2)2=(a−5)2+(b−1)2 for c1(a,b). I expanded and simolified this to get b=3a−9. What do I do next?
The product of the two complex number c₁c₂ is thus 10 + 9i.
Let c₁ = a + ib and c₂ = c + id. Also z₁ represent 2 + 2i and z₂ represent 5 + i. The length of side of the equilateral triangle formed can be obtained by
|z₁ - z₂| = r = √((5 - 2)² + (1 - 2)² = √10
a² + b² - 4a - 4b + 8 = 10
a² + b² - 10a - 2b + 26 = 10
So 6a - 2b = 18 that is 3a - b = 9
Thus b = 3a - 9
Similarly c² + d² - 4c - 4d + 8 = 10
c² + d² - 10c - 2d + 26 = 10
So 3c - d = 9, that is d = 3c - 9
Midpoint of line joining c₁ and c₂ is same as that of z₁ and z₂, so
(c + a)/2 = ( 5 + 2)/ 2 and (d + b)/2 = (1 + 2)/2
So midpoint is 3.5 + 1.5i
Altitude of equilateral triangle of side a is √3a/2
That is √(a - 3.5)² + (b - 1.5)² = √3×√10/2
√(a - 3.5)² + (3a - 9 - 1.5)² = √30 /2
a² - 7a + 12.25 + 9a² - 63a + 110.25 = 30/4
On solving a = (7 ± √3)/ 2
So a = (7 - √3)/ 2, c = (7 + √3)/ 2
b = 3a - 9 = (3 - 3√3)/ 2, d = (3 + 3√3)/2
So c
c₁ = (7 - √3)/2 + i (3 - 3√3)/ 2
c₂ = (7 + √3)/2 + i (3 + 3√3)/ 2
So the product c₁c₂ is
= ((7 - √3)/ 2) * ((7 + √3)/ 2) - ((3 - √3)/ 2) * ((3 + √3)/ 2) + i (((7 - √3)/ 2) * ((3 + √3)/ 2) + ((7 + √3)/ 2) * ((3 - √3)/ 2))
= 10 + 9i
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A type of green paint is made by mixing 2 cups of yellow with 3.5 cups of blue. Find a mixture that will make the different shade of green but has a smaller amount.
Use 1.75 cups of blue and 1 cup of yellow
What is a mixture?
Mixture can be defined as a substance made by combining two or more different materials in such a way that no chemical reaction occurs. A mixture can usually be separated back into its original components.
The given problem can be placed in the category of ratios and proportions.
There is a ratio of color mixing which contains the proportion of two colors i.e blue and yellow. When we use 2 cups of yellow with 3.5 cups of yellow then we get green color so if we mix half of their amounts then we can get less or simply half amount of color too.
Therefore adding 1 cup of yellow and 1.75 cup of blue will give us small amount in result.
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Please help me find each measure!! Thank you
find the domain and range of the function. Use a graphing utility to verify your results. (Enter your answer using interval notation.)
f(x) = ?x2 ? 6x + 7
The domain is [0,100].[0,100]. The range is [0,1500] [0,1500]
(a) To find the cost of making 25 items substitute
x=25 in the equation
=10+500(25)
=10(25)+500(25)=750
c(x)=10x+500
c(25)=10(25)+500
c(25)=750
the cost of making 25 items is
$750
(b)
Since the maximum cost allowed is
$1500
10+500≤1500
10x+500≤1500
To solve this inequality
First, subtract 500 from both sides
10≤1000
10x≤1000
Divide both sides by 10
≤100x≤100
This means you can make at most 100 items.
The domain is
[0,100].[0,100].
The range is
[0,1500] [0,1500].
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College Algebra! Worth 20 points. Can anyone help?
The solution set to inequality - 5 ≤ (1 / 2) · (2 · m + 2) ≤ 14 is equal to - 6 ≤ m ≤ 13. (Correct choice: A)
How to solve a simultaneous inequality
Herein we find a simultaneous inequality, that is, the combination of two inequalities. We find the following case below by algebra properties:
- 5 ≤ (1 / 2) · (2 · m + 2) ≤ 14
(1 / 2) · (2 · m + 2) ≥ - 5 and (1 / 2) · (2 · m + 2) ≤ 14
Now we proceed to determine the solution set to this inequality by algebra properties:
2 · m + 2 ≥ - 10 and 2 · m + 2 ≤ 28
2 · m ≥ - 12 and 2 · m ≤ 26
m ≥ - 6 and m ≤ 13
- 6 ≤ m ≤ 13
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THIS IS URGENT, What is the GCF of 36x^2 and 18xy^2
The greatest common factor (GCF) of two or more expressions is the largest monomial that divides each expression exactly.
To find the GCF of 36x^2 and 18xy^2, we can factor out the greatest common factor from each expression.
The prime factorization of 36 is 2^2 * 3^2 and of 18 is 2*3^2
The GCF of 36x^2 and 18xy^2 is 18x^2y^2 = 2*3^2 * x^2 * y^2
So, the GCF of 36x^2 and 18xy^2 is 18x^2y^2
Answer:
18xy
Step-by-step explanation:
Read the following prompt and type your response in the space provided.
Give an example of a problem involving multiplication of fractions that can be made easier using the associative property. Explain how it makes the problem easier.
An example of a question that can be solved using the associative method is : (2/5) * (3/4) * (5/6)
How to use the associative propertyOne example of a problem involving multiplication of fractions that can be made easier using the associative property is:
(2/5) * (3/4) * (5/6)
Using the associative property of multiplication, we can change the grouping of the fractions to make the problem easier. The associative property states that the way we group the numbers being multiplied does not change the product. For example,
(2/5) * (3/4) * (5/6) = (2/5 * 3/4) * (5/6) = (6/20) * (5/6) = 30/120
This simplification makes it easier to see that the final answer is 1/4.
By using the associative property, we can first multiply (2/5) * (3/4) = (6/20) which is easier to cancel out the common factor and then multiply it with (5/6) = (5/6) * (6/20) = (30/120) = 1/4 which is the final answer
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Can someone please help me
The cost of each shirt is $6 and the cost of each pant for the given condition is $9.
What is elimination method?The elimination approach involves taking one variable out of the system of linear equations by utilising addition or subtraction together with multiplication or division of the variable coefficients.
Let us suppose the cost of shirt = x.
Let us suppose the cost of pant = y.
Given that 3 shirts and 3 pants cost $45, this is represented as:
3x + 3y = 45 (equation 1)
1 shirt and 2 pants cost $24, this is represented as:
x + 2y = 24 (equation 2)
Using the elimination method to solve:
Multiply the second equation with 3 and change the signs of all the variables.
(x + 2y = 24) (3)
-3x - 6y = -72 (equation 3)
3x + 3y = 45
Comparing equation 1 and equation 3, the x term is cancelled, and the remaining equation after subtracting both equations is:
- 3y = -27
y = 9
Substitute the value of y in equation 2.
x + 2y = 24
x + 2(9) = 24
x + 18 = 24
x = 24 -18
x = 6
Hence, the cost of each shirt is $6 and the cost of each pant is $9.
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Calculate the percent by mass of a solution where 13.687 grams is dissolved in 46.833 grams of water. answer without the percent sign
The percent by mass of a solution is 29.21.
To calculate the percent by mass of a solution, divide the mass of the solute (13.687 grams) by the total mass of the solution (46.833 grams). Then, multiply this value by 100 to get the percent.
% by mass = (mass of solute/total mass of solution) x 100
% by mass = (13.687/46.833) x 100 = 29.21.This calculation can be used to determine the amount of a solute in a given solution. It is important to know the percent by mass of a solution in order to make sure the solution is composed of the desired amount of solute. Knowing the percent by mass can also be used to compare the solutions of different concentrations.
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43)A tree13feet high grows at the rate of3feet each year. How many years will it take for the tree to grow to a height of 28 feet?
the answer is 5 please show me how to get that answer?
A 5 foot wide painting should be centered on a 13 foot wall type the left side of an equation that can be used to determine how many feet should be on each side of the painting
The number of feet that should be on each side of the painting is 4 feet.
How to illustrate the equation?An equation simply has to do with the statement that illustrates the variables given. In this case, it is vital to note that two or more components are considered in order to be able to describe the scenario.
Let x be the number of feet on each side of the painting.
The equation to determine the number of feet on each side of the painting would be:
x + 5 + x = 13
So x = (13-5)/2
x = 4
So there should be 4 feet on each side of the painting.
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Si el costo de 1 Kg de azucar es $8?cual es el precio de 3 3/4 kg?
Answer:
$30
Step-by-step explanation:
1kg= $8
3kg *$8= $24
3/4= 0.75
$8 *0.75= $6
$24+$6= $30
3.75 kg * $8= 30
* = multiply
in exercises 11 and 12, determine if b is a linear combination of subscript[a, 1], subscript[a, 2], and subscript[a, 3].
Yes b can be written as linear combination of 3 vectors
What is linear combination of vector?A linear combination of vectors is a sum of scaled vectors, where the scaling factors are scalars. In other words, it's a linear combination of the form a_1v_1 + a_2v_2 + ... + a_n*v_n, where v_1, v_2, ..., v_n are vectors and a_1, a_2, ..., a_n are scalars. This is a fundamental concept in linear algebra and vector spaces and is used in many areas of mathematics and physics.
if b is linear combination of [tex]a_1,a_2,a_3[/tex]
then [tex]xa_1+ya_2+za_3=b[/tex]
from here we got :
x+5z = 2
-2x+y-6z = -1
2y + 8z = 6
after solving the above equation we get the solution as
{x,y,x} = {2,3,0} is a solution to the above equation so b is a linear combination of [tex]a_1,a_2,a_3[/tex]
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Complete question:
determine if b is a linear combination of [tex]a_1,a_2,a_3[/tex]
[tex]a_1[/tex] = [tex]\left[\begin{array}{ccc}1\\-2\\0\end{array}\right][/tex] , [tex]a_2 = \left[\begin{array}{ccc}0\\1\\2\end{array}\right][/tex] , [tex]a_3 = \left[\begin{array}{ccc}5\\-6\\8\end{array}\right][/tex] , b= [tex]\left[\begin{array}{ccc}2\\-1\\6\end{array}\right][/tex]