4.02 Lesson Check Arithmetic Sequences (2)

Answer:

Let L be the length of the rectangle.

According to the problem, the width of the rectangle is 4 units less than the length, so the width is L - 4.

The area of the rectangle is given as 32 square units, so we can set up the equation:

A = L(L - 4) = 32

Expanding the left side of the equation, we get:

L^2 - 4L = 32

Rearranging and factoring, we get:

L^2 - 4L - 32 = 0

We can solve for L by using the quadratic formula:

L = (-b ± sqrt(b^2 - 4ac)) / 2a

Where a = 1, b = -4, and c = -32.

L = (-(-4) ± sqrt((-4)^2 - 4(1)(-32))) / 2(1)

L = (4 ± sqrt(16 + 128)) / 2

L = (4 ± sqrt(144)) / 2

L = (4 ± 12) / 2

L = 8 or L = -4

Since the length must be a positive value, we take L = 8.

Therefore, the length of the rectangle is 8 units.

The width and length of the rectangle will be 4 units and 8 units, respectively.

Let W be the rectangle's width and L its length.

The area of the rectangle is the multiplication of the two different sides of the rectangle. Then the rectangle's area will be

Area of the rectangle = L × W square units

The width of a square shape is 4 units less than the length. The region of the square shape is 32 square units. Then the equations are given below.

[tex]W = L - 4[/tex]            ...1

[tex]L \times W = 32[/tex]           ...2

From equations 1 and 2, then we have

[tex]L \times (L - 4) = 32[/tex]

[tex]L^2 - 4L = 32[/tex]

[tex]L^2 - 4L - 32 = 0[/tex]

[tex]L^2 - 8L + 4L - 32 = 0[/tex]

[tex]L(L - 8) + 4(L - 8) = 0[/tex]

[tex](L - 8)(L + 4) = 0[/tex]

[tex]L = 8, -4[/tex]

Then the width of the rectangle is given as,

[tex]W = 8 - 4[/tex]

[tex]W = 4 \ units[/tex]

The width and length of the rectangular shape will be 4 units and 8 units, separately.

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Answer:

Step-by-step explanation:

2

Answer:

145.95

Step-by-step explanation:

I took the test and got it right

Matching each number with its name, 1,254 :: whοle number

Whοle numbers are a set οf numbers including all natural numbers and 0. They are a part οf real numbers that dο nοt include fractiοns, decimals, οr negative numbers. Cοunting numbers are alsο cοnsidered as whοle numbers.

Whοle numbers are pοsitive integers (including zerο) withοut any fractiοnal οr decimal parts. They are used tο cοunt whοle οbjects οr things, and cannοt represent a part οf a whοle. Examples are 0, 1, 2, 3, 4, 5, and sο οn.

0.143143143.... :: repeating decimal

0.123456789..... :: nοn-terminating/nοn-repeating

0.13 :: terminating

1,254 :: whοle number

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The solution to the system of equations is C = 3 and H = 9.

Let C = Cora's miles last week and H = Hana's miles last week

The equation for Cora's miles is C + 7x = 3, and the equation for Hana's miles is H + 4x = 9, where x is the number of weeks.

The solution to this system of equations is C = 3, and H = 9. This can be determined by solving each equation separately and then checking that the answers satisfy both equations.

For Cora's equation, subtract 7x from both sides to get C = 3 - 7x. Solving for x gives x = (3 - C) / 7. Plugging this into Hana's equation gives H + 4((3 - C) / 7) = 9. Simplifying this equation gives H = 9 - 12 + 4C.

Checking the answers, C = 3 - 7(0) = 3, and H = 9 - 12 + 4(3) = 9. Both equations are satisfied by C = 3 and H = 9, so this is the solution to the system of equations.

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Answer:

$53.60

Step-by-step explanation:

This statement is knοwn as the Pythagοrean Theοrem, and it states that the square οf the hypοtenuse οf a right triangle is equal tο the sum οf the squares οf the οther twο sides.

The area is the amοunt οf twο-dimensiοnal space that is enclοsed within a given bοundary οr shape. It is measured in square units, such as square inches, square feet, οr square meters. The area is an impοrtant cοncept in mathematics and is used tο calculate the size οf physical οbjects as well as tο measure figures οn a twο-dimensiοnal surface. Knοwing hοw tο calculate area is an essential skill in a variety οf fields, including architecture, engineering, and landscaping.

This relatiοnship was first discοvered by the ancient Greek philοsοpher Pythagοras οf Samοs, whο nοticed that this equatiοn always held true in right-angled triangles. The equatiοn is expressed as, where a and b are the sides οf the triangle and c is the hypοtenuse.

Fοr example, Lets take an example οf twο legs 3 and 4, and the hypοtenuse be 5,

Lets see if the hypοtenuse² =  leg a² + leg b²

⇒ 5² = 4² + 3²

⇒ 25 = 16 + 9

⇒ 25 = 25.

Thus, This statement is knοwn as the Pythagοrean Theοrem, and it states that the square οf the hypοtenuse οf a right triangle is equal tο the sum οf the squares οf the οther twο sides.

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The complete question is:

Is it correct that the sum of the areas of two squares on the legs (a and b) equals the area of the square on the hypotenuse (c)?

The solution for g is -2

[tex]-g+2\left(3+g\right)=-4\left(g+1\right)[/tex]

expand [tex]-g+2\left(3+g\right)[/tex]

= [tex]g+6[/tex]

expand  [tex]-4(g+1)[/tex]

[tex]-4g-4[/tex]

[tex]g+6=-4g-4[/tex]

collect like terms:

[tex]5g=-10[/tex]

Divide both side by 5:

[tex]\frac{5g}{5} =\frac{-10}{5}[/tex]

=[tex]g = -2[/tex]

Therefore, the solution for g is -2.

A linear equation is an algebraic equation in which the highest power of the variable is 1. It can be written in the form of y = mx + b, where y is the dependent variable, x is the independent variable, m is the slope of the line, and b is the y-intercept.

The slope is the rate of change of the line, which represents how steeply it rises or falls, and the y-intercept is the point where the line intersects the y-axis.

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The area of the triangle with sides 26 feet, 18 feet, and 9 feet is approximately 125.6 square feet.

To find the area of a triangle while given three aspects, we are able to use Heron's formula, which states that the region of a triangle with facets a, b, and c is given by means of:

area = √(s(s-a)(s-b)(s-c))

where s is the semiperimeter, calculated as:

s = (a + b + c) / 2

plugging inside the given values, we get:

s = (26 + 18 + 9) / 2 = 26.5

area = √26.5(26.5-26)(26.5-18)(26.5-9))

area = √(26.50.58.5×17.5)

area = √(15793.75)

area ≈ 125.6 square feet

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Yes, the Median of a triangle is also an altitude and the triangle is isosceles.

The altitude of a triangle is the perpendicular line segment drawn from the vertex of the triangle to the side opposite to it

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side.

In an isosceles triangle, two sides that are equal meet at a vertex that lies directly above the base midpoint. Because of this altitude from the vertex bisect the base at midpoint median to the base from the vertex.

Therefore, Median of a triangle is also an altitude and the triangle and it is isosceles.

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The value of the expression x² + 2y ÷ 2w + 3z for w = 2, x = 5, y = 8, and z = 3 will be; ⇒ 3.15

Mathematical expression is defined as the collection of numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The expression is;

⇒ x² + 2y ÷ 2w + 3z

And, The values are w = 2, x = 5, y = 8, and z = 3.

Now,

The expression is;

⇒ x² + 2y ÷ 2w + 3z

Substitute all the values of w = 2, x = 5, y = 8, and z = 3 in above equation, we get;

⇒ x² + 2y ÷ 2w + 3z

⇒ 5² + 2 × 8 ÷ 2 × 2 + 3 × 3

⇒ 25 + 16 ÷ 4 + 9

⇒ 41 ÷ 13

⇒ 3.15

Thus, The value of the expression x² + 2y ÷ 2w + 3z for w = 2, x = 5, y = 8, and z = 3 will be;

⇒ 3.15

The complete question is-

What is the value of x2 + 2y ÷ 2w + 3z for w = 2, x = 5, y = 8, and z = 3?

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In the Closing Entries, the Income summary, computer service revenue, and retained earnings are given.

A Statement of Income, also known as an Income Statement or Profit and Loss Statement (P&L), is a financial statement that reports a company's revenues, expenses, and net income or net loss over a specific period of time, usually a month, quarter, or year.

The statement shows the company's ability to generate revenues and manage expenses during the period being reported

It is an essential tool for analyzing a company's financial performance and is used by investors, creditors, and management to make decisions about the company's operations.

The journal entries, adjusted balances, closing entries, unadjusted trial balances, and statements of income have all been solved in the images below.

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After addressing the issue at hand, we can state that Keep in mind that any probability model must have these probabilities sum up to 1, as is the case here.

Calculating the probability that an event will occur or that a statement is true is the subject of probability theory in mathematics. A risk is a number between 0 and 1, where 1 denotes certainty and an approximate probability of 0 denotes how likely an event appears to be to occur. A mathematical representation of the likelihood that an event will take place is called probability. You can also express probabilities as percentages ranging from 0% to 100% or as integers between 0 and 1. the proportion of equally plausible options that actually occur when compared to all possible outcomes, leading to a certain event.

A probability model for selecting a bead can be defined as follows:

Let the actions of choosing a glass, wood, or brass bead be represented by G, W, and B, respectively. We can suppose that the likelihood of choosing a particular type of bead is inversely correlated with the quantity of that type of bead in the box. As a result, we have:

P(W) = 96/300 = 8/25 P(B) = 144/300 = 12/25 with P(G) = 60/300 = 1/5

Keep in mind that any probability model must have these probabilities sum up to 1, as is the case here.

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The correct question is -

Write a probability model for choosing a bead.

Choosing Beads

Glass 60

Wood 96

Brass 144

Answer:

y = -1/3x - 5

Step-by-step explanation:

The slope-intercept form is y = mx + b

m = the slope

b = y-intercept

Slope = rise/run or (y2 - y1) / (x2 - x1)

Points (-6, -3) (6, -7)

We see the y decrease by 4, and the x increase by 12, so the slope is

m = -4/12 = -1/3

Y-intercept is located at (0, -5)

So, our equation is y = -1/3x - 5

Answer:

V = 3 * 3 * 7 + 2 * 3 * 5 = 63 + 30 = 93       just adding 2 segments

Answer:

211.55

Step-by-step explanation:

Answer: 215.83mm

Step-by-step explanation: to get the surface area of a cone, you need to plug in the radius (r) and the height (h) to this problem: A=Pi*r(r+{h to the power of 2+r to the power of 2} squared) so the actual problem would be A=Pi*3.7(3.7+{14.4 to the power of 2+ 3.7 to the power of 2} squared)=215.83mm

a. The first term of the sequence is 16.

b. the common difference is d = 32

c. the explicit formula is f(n) = 16 + (n - 1)32

d. the stone falls a distance of 144 feet during the 5th second.

a. The first term of the arithmetic sequence is given to be 16.

b. To find the common difference, we can subtract the second term from the first term, then the third term from the second term:

48 - 16 = 32

80 - 48 = 32

Therefore, the common difference is d = 32.

c. Using f(n) = f(1) + (n - 1)d,

where

f(1) = 16 and d = 32,

we can write the explicit formula for the distance traveled during the nth second as:

f(n) = 16 + (n - 1)32

d. To find the distance traveled during the 5th second, we plug in n = 5 into the explicit formula:

f(5) = 16 + (5 - 1)32

f(5) = 16 + 4*32

f(5) = 16 + 128

f(5) = 144

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Therefore , the solution of the given problem of volume comes out to be  Lake B's volume is roughly as follows is 5.28760675 cubic metres.

The volume of a three-dimensional item, which is measured in cubic units, describes how much room it occupies. Cubic measurements are represented by the symbols litre and in3. However, you can calculate an object's measurements using its bulk. In most cases, the item's weight is transformed into mass units like kilogrammes and kilos.

Here,

=> Lake B's capacity is 2.5 times greater than Lake A's.

=>  Lake B's volume is 2.5 times that of Lake A.

=>  Lake B's volume is 2.5 x 21,150,427,000.

=> Lake B has a volume of 52,876,067,500.

By moving the decimal point until there is only one non-zero digit left of the decimal point, we can represent this number as the sum of a single integer and a power of 10. We need to multiply by 1010 to account for the fact that we shifted it 10 positions to the left.

Therefore, Lake B's volume is roughly as follows:

=> 5.28760675 cubic metres

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Answer:

Proofs attached to answer

Step-by-step explanation:

Proofs attached to answer

The probability that a randomly selected adult from the state has high blood pressure but not high cholesterol is 0.40 or 40%.

To find the probability that a randomly selected adult from the state has high blood pressure but not high cholesterol, we need to subtract the probability of having both high blood pressure and high cholesterol from the probability of having high blood pressure.

Using set notation, let A be the event of having high blood pressure and B be the event of having high cholesterol. Then, the probability of having both high blood pressure and high cholesterol can be written as

P(A ∩ B)= 0.20

The probability of having high blood pressure can be written as

P(A) = 0.60

Therefore, the probability of having high blood pressure but not high cholesterol is:

P(A) - P(A ∩ B) = 0.60 - 0.20 = 0.40

Thus, the probability that a randomly selected adult from the state has high blood pressure but not high cholesterol is 0.40 or 40%.

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Answer:check explanation

Step-by-step explanation:1. √25 is a rational number because it is equal to 5, which is a rational number.

2. 4.31 is a rational number because it can be expressed as the ratio of two integers (431/100).

3. -2/2 is a rational number because it can be simplified to -1, which is a rational number.

4. √18 is an irrational number because it cannot be expressed as the ratio of two integers. It is approximately equal to 4.242640687119285.

5. 7 is a rational number because it can be expressed as the ratio of two integers (7/1).

6. Pi is an irrational number because it cannot be expressed as the ratio of two integers or as a terminating or repeating decimal. It is approximately equal to 3.141592653589793.

7. √50 is an irrational number because it cannot be expressed as the ratio of two integers. It is equal to √(25*2) which simplifies to 5√2.

8. -5 is a rational number because it can be expressed as the ratio of two integers (-5/1).

9. 1/3 is a rational number because it can be expressed as the ratio of two integers (1/3 = 0.33333...).

10. 0 is a rational number because it can be expressed as the ratio of two integers (0/1).

Answer: 24

Step-by-step explanation: 2x2x2 is the volume since 2x2x2 is 8. so the surface area of one side would be 2x2. 4x6= 24. Since there are 6 sides, the answer is 24.

The value of x and y for the rectangle has side lengths of x cm and (x+y) cm are found as : x = 3 and y = 16.

Dimensions are:

Then,

Area = l*w

57 = x(x + y)  ....eq 1

Perimeter = 2(l + w)

44 = 2(x + x + y)

22 = 2x + y

y = 22 - 2x ...put in eq 1

57 = x(x + y)

57 = x(x + 22 - 2x)

57 = x(22 - x)

Simplifying:

57 = 22x - x²

Or.

x² - 22x + 57

Factorizing using quadratic formula:

x = 19 ans x = 3

For x = 19

y = 22 - 2*19

y = 22 - 38  (negative value not possible)

So, x = 3

y = 22 - 2*3

y = 22 - 6

y = 16

Thus, the value of x and y for the rectangle has side lengths of x cm and (x+y) cm are found as : x = 3 and y = 16.

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The coin will reach the stream in 5.61 seconds.

Since the height of the coin, after t seconds, is given by the equation:

h = -16t² + 72t + 100

The stream is the ground level. Thus, the coin reaches the stream when h(t) = 0.

So we can equate the equation to zero and solve for t:

-16t² + 72t + 100 = 0

Using the quadratic formula:

t = −b ± √(b² − 4ac) 2a

where a = -16, b = 72 and c = 100

t = −72 ± [√(72² − 4(-16)(100))] /2(-16)

t =  −72 ± [√11584]/(-32)

t = (-72 ± 107.63)/(-32)

t =  (-72 + 107.63)/(-32) or  (-72 - 107.63)/(-32)

t = -1.11 or  5.61

Since t cannot be negative. Thus, t = 5.61 seconds

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Complete Question

Travis tossed a coin off a bridge into the stream below. The path of the coin can be represented by the equation h = -16t² + 72t + 100. How long will it take the coin to reach the stream?

Angle measure [tex]\frac{23 \pi}{3}[/tex] is associated with the coordinate [tex]\left({\frac{1}{2}}\;-{\frac{\sqrt{3}}{2}}\right)$[/tex]

Angle measure [tex]-\frac{3 \pi}{4}[/tex] is associated with the coordinate [tex]\left(-{\frac{\sqrt{2}}{2}},-{\frac{\sqrt{2}}{2}}\right)[/tex]

Angle measure 150° is associated with the coordinate [tex]\left(-{\frac{\sqrt{3}}{2}},{\frac{1}{2}}\right)$[/tex]

An angle is fοrmed when twο lines οr rays meet at a cοmmοn pοint. The cοmmοn pοint is knοwn as the vertex.

In geοmetry, an angle measure can be defined as the measure οf the angle fοrmed by the twο rays οr arms at a cοmmοn vertex.

Angles are measured in degrees (°) using a prοtractοr. A prοtractοr is a measuring device that is used tο calculate οr draw angles in terms οf degrees.

Check the figure given in the attachment to better understand

Angle measure [tex]\frac{23 \pi}{3}[/tex] is associated with the coordinate [tex]\left({\frac{1}{2}}\;-{\frac{\sqrt{3}}{2}}\right)$[/tex]

Angle measure [tex]-\frac{3 \pi}{4}[/tex] is associated with the coordinate [tex]\left(-{\frac{\sqrt{2}}{2}},-{\frac{\sqrt{2}}{2}}\right)[/tex]

Angle measure 150° is associated with the coordinate [tex]\left(-{\frac{\sqrt{3}}{2}},{\frac{1}{2}}\right)$[/tex]

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The volume of the cylinder is approximately 753.98 cubic centimeters.and the volume of the cone is approximately 600.98 cubic centimeters.

The solutions of the question are given as following :-

1.To find the radius of a cone with height 19 cm and width (diameter) 12 cm, we can use the Pythagorean theorem to relate the height, radius, and slant height of the cone. The slant height is the distance from the vertex of the cone to any point on the circular base, and it is related to the height and radius by the equation:

slant height² = height² + radius²

We know the height is 19 cm, and the diameter (width) is 12 cm, so the radius is half of the diameter, or 6 cm. We can now substitute these values into the equation for the slant height:

slant height²= 19²+ 6²

slant height² = 361 + 36

slant height² = 397

slant height ≈ 19.92 cm

Therefore, the radius of the cone is approximately:

radius = √(slant height² - height²)

radius = √(397 - 19²)

radius ≈ 5.65 cm (rounded to two decimal places)

2.Volume = (1/3) * π * radius²* height

Substituting the given values, we get:

Volume = (1/3) * π * (5.65 cm)²* 19 cm

Volume ≈ 600.98 cubic centimeters

Therefore, the volume of the cone is approximately 600.98 cubic centimeters.

3.To find the volume of the cylinder with height 15 cm and radius (width/2) 4 cm, we can use the formula for the volume of a cylinder:

Volume = π * radius² * height

Substituting the given values, we get:

Volume = π * (4 cm)² * 15 cm

Volume = π * 16 cm² * 15 cm

Volume ≈ 753.98 cubic centimeters

Therefore, the volume of the cylinder is approximately 753.98 cubic centimeters.

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The probability as a function of the sum of independent Poisson distributed random variables. We can then obtain the desired expression for P(X_n = i|X_n−1 = j) as follows:P(X_n = i|X_n−1 = j) = P(Z_1 + ... + Z_j = i) = (1/i!)×(γi)×e{-γ}.

Expression in math is a combination of numbers, variables, operations and symbols that represent a mathematical relationship or value. Expressions can be used to represent equations, functions, and sequences, and can be evaluated for a result. Expressions can include constants, variables, operators, functions, and other mathematical expressions.

The general expression for P(X_n = i|X_n−1 = j) is given by P(X_n = i|X_n−1 = j) = P(Z_1 + ... + Z_j = i), where Z_1, ..., Z_j are independent and identically distributed Poisson random variables with parameter γ. This follows from the fact that the offspring of each individual follows the same Poisson distribution, and the total number of offspring is the sum of the offspring of each individual. Therefore, we can express the probability as a function of the sum of independent Poisson distributed random variables. We can then obtain the desired expression for P(X_n = i|X_n−1 = j) as follows:

P(X_n = i|X_n−1 = j) = P(Z_1 + ... + Z_j = i) = (1/i!)×(γi)×e{-γ}.

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Answer 1:

D. 24 sq feet

The formula to find surface area of a cube is [tex]a=6a^{2}[/tex]

Substitute 2 for a, [tex]2^{2} = 4[/tex]

6 x 4 = 24, so 24 sq feet

Answer 2:

B. 62 sq feet

The formula to find surface area of a rectangular prism is [tex]a = 2(wl+wh+hl)[/tex]

Substitute 3 for w, 5 for l, 2 for h and multiply

a = 62 sq feet

Answer 3:

C. 1296 sq inches

The formula to find surface area of a rectangular prism is [tex]a = 2(wl+wh+hl)[/tex]

Substitute 24 for w, 12 for l, 10 for h and multiply

a = 1296 sq inches

the 99% confidence interval estimate for the proportion of boys born to parents using the YSORT method is: (0.7584, 0.8820)

what is an algebraic expression?

An algebraic expression is a mathematical phrase consisting of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.

a. The point estimate of the population proportion of boys born to parents using the YSORT method can be calculated as the ratio of the number of boys to the total number of babies:

239/291 ≈ 0.8202

Therefore, the best point estimate of the population proportion of boys born to parents using the YSORT method is approximately 0.8202.

b. To construct a 99% confidence interval estimate of the proportion of boys born to parents using the YSORT method, we can use the following formula:

p ± z*sqrt(p(1-p)/n)

where p is the point estimate of the population proportion, z* is the critical value from the standard normal distribution for the desired confidence level (99% in this case), and n is the sample size.

Using the values from part (a), we have:

p = 0.8202

n = 291

z* = 2.576 (from the standard normal distribution table for a 99% confidence level)

Plugging in these values, we get:

0.8202 ± 2.576sqrt(0.8202(1-0.8202)/291)

Simplifying, we get:

0.8202 ± 0.0618

c. Based on the results, the YSORT method appears to be effective in increasing the probability of conceiving a boy. The point estimate of 0.8202 is significantly higher than the theoretical probability of 0.5 for a randomly selected baby to be a boy.

Therefore, the 99% confidence interval estimate for the proportion of boys born to parents using the YSORT method is: (0.7584, 0.8820)

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