(a) find the value of x that maximizes the area of the
figure and (b) find the maximum area.

x in.
x in.
1.5x in.
6 in.

Without using a calculator, the only solution to the equation    [tex](x^3 - 1000)^{1/2} = (x^2 + 100)^{1/3}[/tex] is 10.1136.

To solve this equation without using a calculator, we need to simplify both sides of the equation and then use algebraic techniques to isolate x.

[tex](x^3 - 1000)^{1/2} = (x^2 + 100)^{1/3}[/tex]

square both sides of the equation and cube both sides of the equation, we will have;

(x³ - 1000)³ = (x² + 100)²

We can simplify the left-hand side of the equation by applying the cube of a binomial formula, which states that:

(a + b)³ = a³ + 3a²b + 3ab² + b³

Let's apply this formula with a = x³ and b = -1000:

(x³ - 1000)³ = x⁹ - 3x⁶(1000) + 3x³(1000)² - 1000³

Next, let's simplify the right-hand side of the equation:

(x² + 100)² = x⁴ + 200x² + 10000

Now we can substitute these expressions back into the original equation:

x⁹ - 3x⁶(1000) + 3x³(1000)² - 1000³ = x⁴ + 200x² + 10000

We can then rearrange the terms to get a polynomial equation in x:

x⁹ - 3x⁶(1000) + 3x³(1000)² - x⁴ - 200x² - 10000 - 1000³ = 0

This equation is difficult to solve exactly, but we can make an educated guess that x is close to 10. If we substitute x = 10, we get:

(10³ - 1000)³ ≠ (10² + 100)²

Increase the value of x a little, say 10.1136

(10.1136³ - 1000)³ ≈ (10.1136² + 100)²

This is true, so x = 10.1136 is a solution to the equation. We can check that there are no other integer solutions by noting that the left-hand side of the equation is always larger than the right-hand side for x > 10, and smaller than the right-hand side for x < 10.

Therefore, the only solution to the equation is x = 10.1136.

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Term οr phrase used tο describe the factοrs Geοrge is "quality οf life."

In grammar, a phrase—called expressiοn in sοme cοntexts—is a grοup οf wοrds οr singular wοrd acting as a grammatical unit. Fοr instance, the English expressiοn "the very happy squirrel" is a nοun phrase which cοntains the adjective phrase "very happy".

Phrases can cοnsist οf a single wοrd οr a cοmplete sentence. In theοretical linguistics, phrases are οften analysed as units οf syntactic structure such as a cοnstituent. An adjective phrase is a phrase that mοdifies (explains) a nοun.

The term used tο describe the factοrs Geοrge is cοnsidering is "quality οf life."

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Answer:

??

Step-by-step explanation:

is there an image or something?

Step-by-step explanation:

the 2 angles are supplementary. that means together they have 180°

4x + 29 = 180 - (12x + 55) = 180 - 12x - 55 = 125 - 12x

16x = 96

x = 96/16 = 6

Answer:

(a) Let X be the number of boxes that do not have any balls. We can calculate the probability that a particular box is empty, which is (1-p)^10, where p is the probability that a ball is put into that box. The expected number of boxes that are empty is then:

E(X) = 5(1-p)^10 (since each box has the same probability of being empty)

Substituting ∑(i = 1 to 5) p = 1, we get p = 1/5. Therefore,

E(X) = 5(1-1/5)^10 = 5(4/5)^10 ≈ 0.328

So, the expected number of boxes that do not have any balls is approximately 0.328.

(b) Let Y be the number of boxes that have exactly one ball. The probability that a particular box has exactly one ball is given by:

P(exactly one ball in a box) = 10p(1-p)^4

where p is the probability that a ball is put into that box. The expected number of boxes that have exactly one ball is then:

E(Y) = 5(10p(1-p)^4)

Substituting p = 1/5, we get:

E(Y) = 5(10/5^5) = 1/25

So, the expected number of boxes that have exactly one ball is 1/25.

The correct option is (b) i.e. The cosecant of an angle is defined as the length of the hypotenuse over the length of the side opposite.

What is Trigonometric ratio?

Trigonometric ratios are mathematical functions used in trigonometry, which is the branch of mathematics that deals with the relationships between the sides and angles of triangles. The three main trigonometric ratios are sine, cosine, and tangent

All statements contain some sort of mistake except the statement (b).

The statements should be corrected as given below:

A). The cosine of an angle is defined as the length of the side adjacent over the length of the hypotenuse.

C). The cotangent function is defined as the length of the side adjacent over the length of the side opposite.

D). The sine function is defined as the length of the side opposite over the length of the hypotenuse.

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Using the formula for volume of cylinder we can find that the cup can hold 235.5cm³ volume of water.

The density or quantity of space a cylinder occupies is determined by its volume. Finding a cylinder's volume allows us to determine how much water is required for it to fill up.

Volume of a cylinder equals circle's area times height.

Volume equals πr² h

V = πr²h cubic units, where h is the height and r is the radius, is the volume of a cylinder.

Let's assume the cup is a cylinder. The pencil length is simulating a diagonal (hypothenuse), which forms a right triangle with the height of the cylinder and the diameter at the bottom. So, we apply Pythagorean's Theorem.

13² = 12² + d²

⇒ d² = 169-144

⇒ d = √25

⇒ d =5cm.

Radius = d/2

= 5/2

=2.5cm.

Now,

Volume of the cup = π × r² × h

= 3.14 × 2.5² × 12

= 235.5cm³.

Therefore, the cup can hold 235.5cm³ volume of water.

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The complete question is:

Elena wonders how much water it would take to fill her cup. She drops her pencil in her cup and notices that it just fits diagonally. (See the diagram.) The pencil is 13 cm long and the cup is 12 cm tall. How much water can the cup hold? Explain or show your reasoning.

Therefore , the solution of the given problem of area comes out to be  a.720 sq. ft, b.$227.94 , c.64,800 BTUs. and d.2,592 sq in.

Calculating how much space would be needed to fully enclose its exterior will reveal its overall size. The neighboring area is considered when selecting a comparable good for the rectangular design. The surface area of something determines its overall measurements. The number of sides between a cuboid's four trapezoidal shapes determines how much water it can hold.

Here,

A.

=>  2(144 sq. ft.) + 2(162 sq. ft.) =  720 sq. ft.

b.

16 *  18 = 288 square feet

The entire region that needs painting is:

=> 720 square feet +  288 square feet = 1008 square feet.

We need to apply two layers of paint to the following surface area:

=>  2 * 1008 square feet =  2016 square feet.

5.04 gallons are obtained by multiplying 2016 square feet by 400 square feet per gallon.

=> $6 liters* $37.99/gallon = $227.94

c. 16 feet by 18 feet by 9 feet equals 2,592 cubic feet

As a result, the room's necessary cooling capacity is:

=> 2,592 cubic feet*25 BTUs per cubic foot =  64,800 BTUs.

Rounding up to the next greatest thousand BTUs, we need an air conditioner with a capacity of 65,000 BTUs.

d.

=> 16 ft x 144 sq in/ft = 2,304 sq in by

=>  18 ft x 144 sq in/ft = 2,592 sq in

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Note that in the above Theoretical Probability prompt, the P(A, D) if two cards are randomly selected with replacement is 2/9. (Option C)

Theoretical probability is a branch of mathematics that deals with predicting the likelihood of events based on mathematical reasoning and assumptions, rather than empirical evidence or observations.

Where it is required to determine the P(A, D) if two cards are randomly selected from cards (A,D,D) with replacement:

P (A) = 1/3

P (D) = 2/3


Thus, P (A, D) = 1/3 * 2/3

= 1/3 × 2/3

= (1×2) / (3×3)

= 2/9

Thus, it is correct to state that the probability, if two cards A, and D are randomly selected with replacement is 2/9. (Option C)

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Answer:

Step-by-step explanation:

The range of the uniform distribution is from 23 to 47, so the minimum value (a) is 23 and the maximum value (b) is 47.

The probability density function for a uniform distribution is:

f(x) = 1 / (b - a) if a ≤ x ≤ b

= 0 otherwise

We want to find the probability that a random vehicle gets between 28 and 36 miles per gallon. This is the same as finding the area under the probability density function between x = 28 and x = 36.

Since the distribution is uniform, the probability density function is a horizontal line between x = 23 and x = 47, with height equal to 1 / (47 - 23) = 1/24.

The area under the probability density function between x = 28 and x = 36 is:

P(28 ≤ x ≤ 36) = (36 - 28) * 1/24 = 8/24 = 1/3

Therefore, the probability that a random vehicle gets between 28 and 36 miles per gallon is 1/3, or 0.3333 rounded to four decimal places.

Answer:[tex]-\frac{1}{2}[/tex]

Step-by-step explanation:The line is diagonal to the left(\) , so the slope is negative.

[tex]\frac{rise}{run}[/tex]

Rise one(up), Run two(left)

So the CORRECT answer is [tex]-\frac{1}{2}[/tex]

The domain and the range of the relations are

What is domain?

In mathematics, the domain of a function refers to the set of all possible input values (also known as the independent variable) for which the function is defined.

In other words, the domain of a function is the set of values that can be plugged into the function and produce a valid output.

What is range?

In mathematics, the range of a function refers to the set of all possible output values (also known as the dependent variable) that the function can produce for a given input value from its domain.

In other words, the range of a function is the set of values that the function takes on as its input variable varies over its domain.

Using the above definitions, we have the domain and the range of the relations to be:

Graph 9:

Domain = [0, 6]

Range = [-3, 3]

Relation = Yes

Function = No

Graph 10:

Domain = (-∝, ∝)

Range = (-∝, ∝)

Relation = Yes

Function = Yes

Graph 11:

Domain = (-∝, ∝)

Range = [0, ∝)

Relation = Yes

Function = Yes

Graph 12:

Domain = [-3, 3]

Range = [-4, 4]

Relation = Yes

Function = No

Graph 13:

Domain = (-∝, ∝)

Range = [-2, 4]

Relation = Yes

Function = Yes

Graph 13:

Domain = (-∝, ∝)

Range: y = 1

Relation = Yes

Function = No

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The order of the angles is 43.8°, 63.2, and 75.02 (Here the values are approximate values).

To order the angles of a triangle from least to greatest, we need to first determine the angles of the triangle, and then order the remaining two angles in increasing order.

The formulas we used are

Cosine formula:  Cos C = (a² + b² - c²)/2ab

Law of sine:     a/ sin(A) = b / sin(B) = c / sin(C)

Here we have

The sides of the triangle are 21m 24m and 17m  

Use the Law of Cosines to find one of the angles

=> Cos C = (a² + b² - c²)/2ab

Where a, b, and c are the lengths of the sides of the triangle, and C is the angle opposite the side of length c.

In this case, we have:

a = 21 m, b = 24 m and c = 17 m

Cos(C) = (21² + 24² - 17²) / (2 × 21 × 24)

Cos (C) = 728/1008 = 0.722  

C = cos⁻¹(0.722) = 43.8°

So angle C is 43.8°, which is the smallest angle in the triangle.

To order the remaining two angles in increasing order use the Law of Sines to do this:

=> 21/sin(A) = 24 / sin(B) = 17 / sin(43.78)

=> 21/sin(A) = 24 / sin(B) = 17 /0.69

=> 21/sin(A) = 17/0.69  

=> Sin A = (21 × 0.69)/17

=> Sin A = 0.85

=> A = Sin⁻¹(0.85)

=> A = 63.2

=> 24 / sin(B) = 17 /0.69

=> Sin B = (24 × 0.69)/17

=> Sin B = 0.97

=> B = Sin⁻¹(0.97)

=> B = 75.02

Hence, the order of the angles is 43.8°, 63.2, and 75.02

Therefore,

The order of the angles is 43.8°, 63.2, and 75.02 (Here the values are approximate values).

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Complete Question:

Order the angles from least to greatest 21m 24m 17m

Answer:

Step-by-step explanation:

From the symmetry in the table, we can see that the vertex of the parabola is located at (-5, -2). We will pick another point from the table to use in our model to solve for the a value, just in case there is one. I picked (-4, -1). Any point from the table will work.

Our model is

[tex]y=a(x-h)^2+k[/tex] where h and k are the coordinates of the vertex and x and y are the coordinates from the other point chosen from the table. Filling in to solve for a:

[tex]-1=a(-4+5)^2-2[/tex] and

[tex]-1=a(1)^2-2[/tex] and

-1 = a - 2 so

a = 1.

Now we can fill in the equation with the coordinates of the vertex and the value found for a to get

[tex]y=(x+5)^2-2[/tex] You don't need to put the 1 out in front; it's unnecessary. Your choice is the last one there in the list.

Answer:

36 square inches.

Step-by-step explanation:

volume is given by 1 by 3, into base into height is based. His height and volume and area of the base is given by area of the ashe area of the base is given by 3 into 60 by 5. As given in the question on solving it, we get it equals to 36 square inches.

Answer:

BD = 8

Step-by-step explanation:

To solve this, you need to use Pythagorean theorem which states that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse.

For △ABC, AC (4+16) is the hypotenuse

=> AC^2 = AB^2 + BC^2

For △ABD, AB is the hypotenuse

=> AB^2 = AD^2 + BD^2

For △BCD, BC is the hypotenuse

=> BC^2 = BD^2 + CD^2

Therefore

AC^2 = AB^2 + BC^2

AC^2 = (AD^2 + BD^2) + (BD^2 + CD^2)

20^2 = 4^2 + BD^2 + BD^2 + 16^2

400 - 16 - 256 = 2BD^2

BD^2 = 128/2 = 64

BD = √64 = 8

Answer:

To find the perimeter of the quadrilateral, we need to find the distance between its vertices and then add them up. We can use the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is given by:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Using this formula, we can find the distance between each pair of adjacent vertices:

Distance between (-3, 1) and (2, 1):

d = sqrt((2 - (-3))^2 + (1 - 1)^2) = sqrt(25) = 5

Distance between (2, 1) and (2, -2):

d = sqrt((2 - 2)^2 + (-2 - 1)^2) = sqrt(9) = 3

Distance between (2, -2) and (-3, -2):

d = sqrt((-3 - 2)^2 + (-2 - (-2))^2) = sqrt(25) = 5

Distance between (-3, -2) and (-3, 1):

d = sqrt((-3 - (-3))^2 + (1 - (-2))^2) = sqrt(9) = 3

Now we can add up these distances to get the perimeter:

perimeter = 5 + 3 + 5 + 3 = 16

Therefore, the perimeter of the quadrilateral is 16 units.

Opens down with a vertex at (3, −8)

Answer:

8 < x < 38

Step-by-step explanation:

given 2 sides of a triangle then the third side x is in the range

difference of 2 sides < x < sum of 2 sides , that is

23 - 15 < x < 23 + 15 , then

8 < x < 38

Answer:

Step-by-step explanation:

You want the side lengths in a right triangle geometry when the altitude to the hypotenuse of length 25 cuts off a segment of length 6.

All of the right triangles in this configuration are similar. The proportional relationships between the sides give rise to three geometric mean relations.

In effect each of the segments x, y, z is the geometric mean of the two segments of the hypotenuse it touches.

  x = √(6·(25-6)) = √114 ≈ 10.68

  y = √(6·25) = 5√6 ≈ 12.25

  z = √((25 -6)·25) = 5√19 ≈ 21.79

__

Additional comment

To see how these arise, consider the ratio of long side to hypotenuse:

  z/25 = 19/z

  z² = 25·19

  z = √(25·19) . . . . . the geometric mean of 25 and 19

The "geometric mean" of n numbers is the n-th root of their product. For two numbers, it is the square root of their product.

Using the simple interest formula, we found that the rate of interest at which the interest was paid is 4%.

What is meant by simple interest?

The financial fee for borrowing money is called interest, and it is typically stated as a percentage, such as an annual percentage rate (APR). For the use of their money, lenders may earn interest, and borrowers may pay interest.

A way to figure out how much interest will be charged on a sum of money at a specific rate and for a specific duration of time is by using simple interest. Unlike to compound interest, where the interest from the previous year's principal is added to the current year's principal to determine the interest, the principal under simple interest remains constant.

Given,

The principal amount P = 30 naira

The time period t = 2 years

Interest amount received I = 2.40 naira

We are asked to find the rate of the interest r.

the simple interest formula is:

I = ( P *r * t )/ 100

2.4 = (30*r*2)/100

240 = 60r

r = 240/60  = 4%

Therefore using the simple interest formula, we found that the rate of interest at which the interest was paid is 4%.

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Answer:

shape?Draw a tree diagram to display all the possible outcomes that can occur when you flip a coin and then toss a six-sided die.(b) How many outcomes contain a head and a number greater than 2?(c) Assuming the outcomes displayed in the tree diagram are all equally likely, what is the probability that you will get a head and a number greater than 2 when Sure, I can help you with that! Here's a tree diagram showing all the possible outcomes of flipping a coin and rolling a six-sided die: ``` H T / \ / \ 1 2 3 4 / \ / \ / \ / \ 1 2 3 4 5 6 1 2 ``` Each branch represents a possible outcome, with the first letter indicating the outcome of the coin flip (H for heads, T for tails), and the second number indicating the outcome of the die roll. (b) To find the number of outcomes that contain a head and a number greater than 2, we can look at the branches that start with H and have a 3, 4, 5, or 6 on the second level. There are four of

Answer: There are 3 outcomes that contain greater than 2

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Answer:

Cost per lawnmower = total earnings / number of lawnmowers sold

Cost per lawnmower = €23,380 / 10

Cost per lawnmower = €2,338

Therefore, the cost per lawnmower was €2,338.

Juan paid Rs. 4.35 for the grapes. The solution has been obtained by using the arithmetic operations.

What are arithmetic operations?

The four fundamental operations, frequently referred to as "arithmetic operations," are claimed to be able to explain all real numbers. The four mathematical operations that come after division, multiplication, addition, and subtraction are quotient, product, sum, and difference.

We are given that Juan bought 24 ounces of grapes and the grapes cost 2.90 per pound.

We know that 1 pound = 16 ounces.

So, 24 ounces = 1.5 pounds

Now, using multiplication operation, we get

⇒ 1.5 * 2.9 = Rs. 4.35

Hence, Juan paid Rs. 4.35 for the grapes.

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Anna's monthly payment for principal and interest is $1,117.78.

a) The amount of the required down payment can be calculated as follows:

Down payment = 15% of $245,000

Down payment = 0.15 x $245,000

Down payment = $36,750

Therefore, Anna is required to make a down payment of $36,750.

b) The amount of the mortgage can be calculated as follows:

Mortgage = Total price of the house - Down payment

Mortgage = $245,000 - $36,750

Mortgage = $208,250

Therefore, Anna's mortgage amount is $208,250.

c) The monthly payment for principal and interest can be calculated using the formula for a fixed-rate mortgage:

[tex]$M = P\left[\frac{i(1+i)^n}{(1+i)^n-1}\right][/tex]

Where:

M = Monthly Mortgage

P = Principal (amount of the loan)

i = Monthly interest rate (5% / 12 = 0.0041667)

n = Total number of payments (30 years x 12 months = 360)

Plugging in the values, we get:

[tex]$M = \frac{\$208,250 \left[0.0041667(1+0.0041667)^{360}\right]}{\left[(1+0.0041667)^{360}-1\right]}[/tex]

M = $1,117.78

Therefore, Anna's monthly payment for principal and interest is $1,117.78.

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Therefore, out of the 24 students that Blake interviewed, 9 of them participate in sports.

An equation is a mathematical statement that indicates that two expressions are equal. It typically consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. The expressions on both sides of the equation are separated by an equal sign "=" which means that the two expressions have the same value.

Here,

According to the table, Blake interviewed a total of 24 students. To find out how many of these students participate in sports, we need to add up the number of students who collect sports cards and participate in sports, as well as the number of students who do not collect sports cards and participate in sports. So:

Total students who participate in sports = Number of students who collect sports cards and participate in sports + Number of students who do not collect sports cards and participate in sports

Total students who participate in sports = 6 + 3

Total students who participate in sports = 9

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Answer:

49

Step-by-step explanation:

P: Parenthesis

E: Exponent

M: Multiplication

D: Division

A: Addition

S: Subtraction

Start by solving what is inside the parenthesis first:

Next, solve for the exponent:

7² (which means 7 multiplied by itself twice)

Now you can multiply 49 and 4:

49 × 4 = 196

Finally, divide 196 by 4:

196 ÷ 4 = 49

Therefore, the answer to the expression 7² (4) ÷ 4 is 49.

Answer:

Step-by-step explanation: There are 60 minutes in each hour. You can do 162.44/60 and will get 2 with a remainder of 42.44 and the 42 goes towards minutes will the 0.42 left is for seconds. Your not converting to seconds so we can leave that as a decimal.

This will give you a total of 2 hours 42.44 minutes. If you want to check your work, you can do (2*60)+42.44=162.44.

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To convert 162.44 minutes to hours and minutes, we can follow these steps:

Divide the number of minutes by 60 to get the total number of hours:

162.44 ÷ 60 = 2.7073 hours (rounded to 4 decimal places)

The whole number part of the answer is the number of hours, and the decimal part represents the remaining minutes. In this case, we have:

2 hours and 0.7073 hours (or 42.438 minutes)

To convert the decimal part to minutes, we can multiply it by 60:

0.7073 × 60 = 42.438 minutes (rounded to 3 decimal places)

Finally, we can combine the whole number of hours with the remaining minutes:

2 hours and 42 minutes

Therefore, 162.44 minutes is equal to 2 hours and 42 minutes.

Answer:the top one

Step-by-step explanation: