find the dimensions of a rectangular prism with a fixed surface area of 282 square units and maximum volume. write the exact answer. do not round.

The required volume of the cube is increasing at a rate of 15,000 cubic centimeters per second when the length of each edge is 50 centimeters and the edges are increasing at a rate of 2 centimeters per second.

Let us consider V be the volume of the cube,

And s be the length of one edge of the cube.

Volume of a cube is equal to,

V = s^3

To find how fast the volume is changing with respect to time,

Use the chain rule of differentiation,

dV/dt = dV/ds × ds/dt

where dV/dt is the rate of change of volume with respect to time,

dV/ds is the rate of change of volume with respect to the length of one edge of the cube,

And ds/dt is the rate of change of the length of one edge of the cube with respect to time.

ds/dt = 2 cm/s,

find dV/dt when s = 50 cm.

First, find dV/ds,

dV/ds = 3s^2

Then, we can plug in s = 50 cm,

dV/ds = 3(50)^2

          = 7500 cm^2

Finally, we can plug in the values we have to find dV/dt,

dV/dt = (dV/ds) × (ds/dt)

         = 7500 cm^2/s × 2 cm/s

         = 15,000 cm^3/s

Therefore, the volume of the cube is increasing at a rate of 15,000 cubic centimeters per second .

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The equation is not a proportion, and the value of z in the equation is 17

Given that

2Z = Z + 17

To solve for z in the proportion:

2z = z + 17

We can start by simplifying the equation by subtracting z from both sides:

2z - z = 17

Simplifying further, we get:

z = 17

Therefore, the solution to the proportion is z = 17.

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To solve the problem, we can use the formula for the future value of an annuity:

FV = PMT * ( (1 + r)^n - 1 ) / r

where:

- PMT is the periodic payment

- r is the periodic interest rate

- n is the number of periods

In this case,

- PMT = $475 (the monthly payment)

- r = the monthly interest rate

- n = 24 (the number of months in the agreement)

We first need to calculate the monthly interest rate. We can find this by dividing the yearly interest rate by 12 months:

r = 18% / 12 months = 0.015

Now we can calculate the future value (FV) of the annuity using the above formula:

FV = $475 * ( (1 + 0.015)^24 - 1 ) / 0.015 = $13,237.19

This is the amount that Christopher would owe after 24 months if he made monthly payments of $475.

However, Christopher pays off the loan after 18 months. To find out how much he owes at this point, we can calculate the future value after 18 months:

FV = $475 * ( (1 + 0.015)^18 - 1 ) / 0.015 = $10,198.66

So, after 18 months, Christopher owes approximately $10,198.66.

Answer: SHawn

Step-by-step explanation:

Since there are 52 cards in a standard deck, there are 52 possible outcomes for the first draw from the first deck. Similarly, there are 52 possible outcomes for the first draw from the second, third, and fourth decks. Therefore, the total number of outcomes for the first draw is 52 + 52 + 52 + 52 = 208.

For the second draw, there are only 51 possible outcomes for each deck, since one card has already been drawn. Therefore, the total number of outcomes for the second draw is 51 + 51 + 51 + 51 = 204.

For the third draw, there are only 50 possible outcomes for each deck, since two cards have already been drawn. Therefore, the total number of outcomes for the third draw is 50 + 50 + 50 + 50 = 200.

For the fourth draw, there are only 49 possible outcomes for each deck, since three cards have already been drawn. Therefore, the total number of outcomes for the fourth draw is 49 + 49 + 49 + 49 = 196.

The total number of outcomes for drawing one card each from four different standard decks of cards is the product of the outcomes for each draw: 208 x 204 x 200 x 196 = 338,401,792.

Answer: The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height.

For the original cone with radius r and height 12 cm, we have:

V = (1/3)πr^2h

V = (1/3)πr^2(12)

V = 4πr^2

For the second cone with 3 times the radius of the original cone and the same height of 12 cm, we have:

V' = (1/3)π(3r)^2h

V' = (1/3)π9r^2(12)

V' = 36πr^2

Therefore, the expression for the volume of the second cone is 36πr^2.

Step-by-step explanation:

The IQR is the best measure of variability for this data set, and its value is 4.

What is median?

Median is a measure of central tendency that represents the middle value in a dataset when the values are arranged in order of magnitude.

The correct answer is:

The IQR is the best measure of variability, and it equals 4.

The box plot shows the median (represented by the line in the box), the interquartile range (IQR) (represented by the length of the box), and the range (represented by the lines outside the box).

The IQR is the range between the first quartile (Q1) and the third quartile (Q3) of the data set, which contains the middle 50% of the data. In this case, the box extends from 17 to 21, indicating that Q1 is 17 and Q3 is 21. Therefore, the IQR is the difference between Q3 and Q1, which is 21-17=4.

The range is the difference between the maximum and minimum values in the data set, which is 27-10=17. However, the range can be affected by outliers and is not as robust as the IQR.

Therefore, the IQR is the best measure of variability for this data set, and its value is 4.

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Your chance of being chosen to attend the festival is approximately: Probability = 1 / 23 ≈ 0.0435 or 4.35% when rounded to four decimal places.

To determine your chance of being chosen to attend the festival, you need to find the probability of your name being

drawn from the basket.

There are 22 other people in class, and each person, including you, will have their name on a piece of paper.

Total number of names in the basket = 22 (other people) + 1 (you) = 23

Since each person has an equal chance of being chosen, the probability of your name being drawn is:

Probability = (Number of favorable outcomes) / (Total number of outcomes)

Probability = 1 (your name) / 23 (total names)

So, your chance of being chosen to attend the festival is approximately:

Probability = 1 / 23 ≈ 0.0435 or 4.35% when rounded to four decimal places.

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Answer: 163.6363...

Step-by-step explanation:

44% of c is 72, so then [tex]0.44*c=72\\[/tex].

Divide both sides by 0.44 to solve for c, and you get c=163.6363...

Tip: if you are ever trying to figure out what a number is a percentage of, like in this problem, just divide the number by the percent in decimal form.

Answer:

[tex] \sf \: 6x + 14 [/tex]

Step-by-step explanation:

Now we have to,

→ Simplify the given expression.

The property we use,

→ Distributive property.

The expression is,

→ 2(3x + 7)

Let's simplify the expression,

→ 2(3x + 7)

→ 2(3x) + 2(7)

→ (2 × 3)x + 14

→ 6x + 14

Hence, the answer is 6x + 14.

Naomi walks a total of 6 + 4 + 7.2 ≈ 17.2 meters.

How to calculate?

Naomi walks the length of the room, which is 6 meters, then across the width, which is 4 meters.

Using the Pythagorean theorem, the length of the diagonal can be found:

diagonal = √(6²2 + 4²2)

diagonal = √(36 + 16)

diagonal = √52

diagonal ≈ 7.2 meters

Therefore, Naomi walks a total of 6 + 4 + 7.2 ≈ 17.2 meters.

The Pythagorean theorem is a fundamental concept in mathematics that states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

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Answer:

To find the number of years it will take for the population of the colony to grow from 80,000 to 92,610 with a growth rate of 5% per annum, we can use the formula for compound interest:

Final Population = Initial Population * (1 + Growth Rate) ^ Number of Years

92,610 = 80,000 * (1 + 0.05) ^ t

Now, we can solve for t:

(92,610 / 80,000) = (1.05) ^ t

1.157625 = (1.05) ^ t

To find t, we can use logarithms:

t = log(1.157625) / log(1.05)

t ≈ 2.967

So, it will take approximately 2.967 years for the population to grow from 80,000 to 92,610 at a 5% growth rate.

Now, let's consider a 2% lower growth rate (5% - 2% = 3%). We can use the same formula to find the final population after the same time (2.967 years):

Final Population = Initial Population * (1 + Growth Rate) ^ Number of Years

Final Population = 80,000 * (1 + 0.03) ^ 2.967

Final Population ≈ 80,000 * 1.09364

Final Population ≈ 87,491.2

To find the difference in population for the same time, we can subtract the population with the lower growth rate from the population with the higher growth rate:

Difference in population = 92,610 - 87,491.2

Difference in population ≈ 5,118.8

So, the difference in population for the same time with a 2% lower growth rate would be approximately 5,118.8.

Answer:

O=0

T=-5

c = 0

R = -2

6 = 8

2 = 8

11=-3

Step-by-step explanation: Sorry for the late answer It took me a bit to solve all the problems but I hope this helps you!!

The resulting triangle would have twice the area and all sides would be twice as long as the original triangle, but the angles would remain the same.

A triangle is a geometric shape that consists of three straight line segments or sides that are connected at three points called vertices. Triangles are two-dimensional and are the simplest polygon that can exist in Euclidean space. Triangles can be classified according to the size and shape of their sides and angles. Triangles are used in many different areas of mathematics, science, and engineering. They are also common in everyday life, for example in construction, architecture, and design.

Here,

To dilate a triangle with a center point C and a scale factor of 2, you would first draw a ray from C through each vertex of the triangle. Then, you would measure the distance from C to each vertex of the original triangle and multiply it by the scale factor (2 in this case) to find the distance from C to each corresponding vertex of the dilated triangle. Finally, you would draw lines connecting the dilated vertices to form the dilated triangle.

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Answer: 200 teachers attended the conference

Step-by-step explanation:

        To answer this question, we will find 5% of 4,000. A percent divided by 100 becomes a decimal, and "of" means multiplication in mathematics.

        5% / 100 = 0.05

        4,000 * 0.05 = 200 teachers attended the conference

The answer is 200

If 5% of teachers at a university attended a conference and 4000 teachers are enrolled at the university, about 200 teachers attended the conference. What is the problem asking for? The problem is asking us to calculate how many teachers attended a conference given that 5% of the total number of teachers at a university attended the conference and that the university has 4000 teachers .What is the formula for percentage? The formula for percentage is given by:(Part/Whole)*100Let T be the total number of teachers at the university, and let x be the number of teachers that attended the conference. We know that 5% of teachers at a university attended a conference, therefore:(5/100)*T = x,. Given that there are 4000 teachers in the university, we can substitute this value into the equation: (5/100)*4000 = x. Simplifying the equation, we get: x = 0.05 * 4000x = 200.

Therefore, about 200 teachers attended the conference.

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Answer:

6x² - 12x - 18

the factors are -18 and 6

6x² - 18x + 6x - 18

6x(x - 3) + 6 (x - 3)

(6x + 6) or (x - 3)

the factors are:

(6x + 6) or (x - 3)

Answer:

Graph H

Step-by-step explanation:

intercept (0,2)

Answer:

(6-3)

Step-by-step explanation:

From the given percentage values, 326.25 points come from amount owed and length of credit history.

What is percentage?

Percentage is a method of representing a value as a fraction of 100. This is frequently used to make comparisons, demonstrate ratios, or determine a fraction of a whole.

The borrower's credit score is 725, and we need to find out how many points come from amount owed and length of credit history.

From the information given, we know that these two categories make up a total of 45% of the score calculation (30% for amount owed and 15% for length of credit history).

We can set up a proportion:

(30% + 15%) / 100% = x / 725

Simplifying the left side:

45% / 100% = 0.45

0.45 = x / 725

Multiplying both sides by 725:

x = 0.45 * 725

x = 326.25

Therefore, 326.25 points come from amount owed and length of credit history.

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Answer:

aₙ = 3n - 2

Step-by-step explanation:

The given sequence is an arithmetic sequence with the first term a₁ = 1 and the common difference d = 3.

The explicit formula for an arithmetic sequence is:

aₙ = a₁ + (n - 1)d

where aₙ represents the nth term in the sequence.

Substituting the values we have:

aₙ = 1 + (n - 1)3

Simplifying the equation we get:

aₙ = 3n - 2

Therefore, the explicit formula for the given sequence 1, 4, 7, 10, ... is aₙ = 3n - 2.

Answer:

[tex] - \frac{2}{5} m( - \frac{4}{9} m - \frac{1}{7} ) = [/tex]

[tex] \frac{8}{45} {m}^{2} + \frac{2}{35} [/tex]

Answer:

a = 9 , b = 5 , c = 25

Step-by-step explanation:

using the Cosine rule in the triangle

a² = b² + c² - 2bc cosA

where a is the side opposite ∠ A and b, c the sides adjacent to ∠ A

here

a = y , b = x + 5 , c = 3x and A = Θ , then

y² = (x + 5)² + (3x)² - [ 2(x + 5) × 3x × cosΘ ]

y² = x² + 10x + 25 + 9x² - [ 6x(x + 5) × [tex]\frac{1}{6}[/tex] ]

y² = 10x² + 10x + 25 - x(x + 5)

y² = 10x² + 10x + 25 - x² - 5x

y² = 9x² + 5x + 25 ← in the form y² = ax² + bx + c

with a = 9 , b = 5 , c = 25

Therefore, the general form of the circle is:

[tex]x^2 + y^2 - 10x-10y+34=0[/tex].

A circle is a geometric shape that is defined as the set of all points in a plane that are at a fixed distance (called the radius) from a given point (called the center). The distance from the center to any point on the circle is always the same. Circles are often represented using the symbol "O" or "⚪" and are important in mathematics, geometry, and physics. They have many properties, such as circumference (the distance around the circle), area (the space inside the circle), and diameter (the distance across the circle through the center). Circles are used in various fields, including architecture, engineering, and art.

To convert the standard form of a circle to the general form, we expand and simplify the equation as follows:

[tex](x - 5)^2 + (y - 5)^2 = 16[/tex]

[tex]x^2 - 10x + 25 + y^2 - 10y + 25 = 16[/tex] (using the identity [tex](a - b)^2 = a^2 - 2ab + b^2)[/tex]

[tex]x^2 + y^2 - 10x-10y+34=0[/tex]

Therefore, the general form of the circle is:

[tex]x^2 + y^2 - 10x-10y+34=0[/tex]

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When a study sample is representative of the target population, it allows for the results to be generalizable, meaning they can be applied to the broader population.

This is an important aspect of research, as it ensures the study's findings are relevant and useful beyond the specific sample studied.

When a study sample shows representativeness, it means that the sample accurately reflects the characteristics of the target population.

In this case, the study is most likely generalizable, with results that can be applied to the target population.
To achieve representativeness, researchers often use random sampling techniques, which help minimize the potential for sampling bias.

Sampling bias occurs when certain groups within the target population are over- or under-represented in the sample, which can lead to inaccurate conclusions.

In contrast, if a study used strict inclusion and exclusion criteria to limit the sample size or relied on specific inclusion criteria, it could result in a sample that is not representative of the target population.

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The three numbers are:

(-1 + √(97))/2, (-1 ± √(97))/2, (-1 ± 7√(97))/2

What is an arithmatic progression?

An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the preceding term. The fixed constant is called the common difference, denoted by d.

Let the three numbers in the GP be a/r, a, and ar, where a is the middle term.

Then we have:

a/r + a + ar = 26 (sum of GP)

(a/r + 5) + (a + 9) + (ar + 5) = 3(a + 4) (sum of AP)

Simplifying the second equation, we get:

a(r - 1) = 3

Substituting this into the first equation, we get:

a/r + a + ar = 26

1/r + 1 + r = 26/a

r^2 + r - 26/a = 0

Solving for r using the quadratic formula, we get:

r = (-1 ± √(1 + 104/a))/2

Since r > 0 (because the terms are in GP), we take the positive root:

r = (-1 + √(1 + 104/a))/2

Substituting this into the equation a(r - 1) = 3, we get:

a(-1 + √(1 + 104/a))/2 - a = 3

-a + a √(1 + 104/a) - 2 = 0

a √(1 + 104/a) = -a + 2

a² (1 + 104/a) = a² - 4a + 4

104a = 4a² - 4a + 4

a² - a + 1 = 26

(a - 1/2)² + 3/4 = 26

(a - 1/2)² = 97/4

a - 1/2 = ±√(97)/2

a = (1 ± √(97))/2

Since the terms are in GP, we can use the value of a to find the other two terms:

a/r = (1 ± √(97))/(2(-1 + √(1 + 104/a))/2) = (-1 ± √(97))/2

ar = (1 ± √(97))/(2(-1 - √(1 + 104/a))/2) = (-1 ± 7√(97))/2

Therefore, the three numbers are:

(-1 + √(97))/2, (-1 ± √(97))/2, (-1 ± 7√(97))/2

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Answer:

No, you cannot simplify it.

Step-by-step explanation:

Because 2 and 3 are one of the lowest numbers that cannot be simplified.

Answer:

PQ = 165.60m

Step-by-step explanation:

Since triangles are similar

PQ/70 + 210 = 124.2/210

PQ = 124.2 (70 + 210)/210 = 165.60m

The distance between the points H (-9, 1), and K (-1, 8) in the coordinate plane, obtained using the distance formula is; Distance: 10.63

The distance formula is a formula that is used to find the distance between two points on the coordinate plane.

The specified points are; H = (-9, 1), and K = (-1, 8)

The distance formula can be used to calculate the distance between the points as follows;

The distance formula is; d = √((x₂ - x₁)² + (y₂ - y₁)²)

Where; (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.

When, (x₁, y₁) = (-9, 1), and (x₂, y₂) = (-1, 8), we get;

d = √((-1 - (-9))² + (8 - 1)²) = √(8² + 7²) =

√(8² + 7²) = √(64 + 49) = √(113) ≈ 10.63 (rounded to the nearest hundredth)

The distance between the points is about 10.63 units

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Therefore , the solution of the given problem of trigonometry comes out to be -1 is the simplified formula.

Some claim that the fusion of various fields contributed to the development of astrophysics. With the aid of precise mathematical methods, many metric issues can be resolved or the output of a computation can be determined. The scientific investigation of all six fundamental geometric calculations is known as trigonometry. They are known by a number of names and abbreviations, such as sine, variance, angle, and others. (csc).

Here,

Let's first use trigonometric identities to separately simplify each term:

=> cos(x - 90°) equals sin(x)

=>  cos(x - 180°) = -cos(x)

=> tan(x) = cos(x) / sin(x)(x)

=>  x(cos(-)) = x(x)

=> (180° + x) sin = -sin(x)

When we replace the equation with these simplifications, we get:

=> cos(x) * sin(180° plus x) * cos(x) * (-cos(x)) + (sin(x) / cos(x))

=>  sin(180° plus x) * sin(-cos2(x))

=>  -cos²(x) - sin(x)*sin(180° + x) (using the equation sin(x)*sin(x)) = -sin^2(x))

Using the identity sin2(x) +cos2(x) = 1, we get = -1.

Consequently, -1 is the simplified formula.

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Answer:

Step-by-step explanation:

The math used shows how much cups it would take to make 4 batches, 3 cups. Therefor yes, he does have exactly enough to make 4 batches of muffins.