The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials. (Let x, y, and z be the dimensions of the aquarium. Enter your answer in terms of V.)

Answers

Answer 1

Answer:

The dimensions of the aquarium that minimize the cost of the materials:

[tex]x=y=\sqrt[3]{\frac{2V}{5}}\\z=\sqrt[3]{\frac{25V}{4}}[/tex]

Step-by-step explanation:

Let x, y and z be the dimensions of aquarium .

Surface area of an aquarium = xy+2yz+2xz

Volume of aquarium V=[tex]Length \times Breadth \times Height=xyz[/tex]  ----A

We are given that slate costs five times as much (per unit area) as glass

So, Cost function : C=5xy+2yz+2xz

Now we will use langrage multiplier to find the dimensions of the aquarium that minimize the cost of the materials.

[tex]\nabla C =\lambda \nabla V[/tex]

[tex](\frac{\partial C}{\partial x},\frac{\partial C}{\partial y},\frac{\partial C}{\partial z})= \lambda (\frac{\partial V}{\partial x},\frac{\partial V}{\partial y},\frac{\partial V}{\partial z})[/tex]

[tex](5y+2z,5x+2z,2y+2x)=\lambda(yz,xz,xy)[/tex]

So,

[tex]5y+2z=\lambda yz[/tex]   ----1

[tex]5x+2z=\lambda xz[/tex] -----2

[tex]2y+2x=\lambda xy[/tex]  ----3

Multiply 1 ,2 and 3 by x,y and z respectively.

[tex]5xy+2xz=\lambda xyz[/tex]   ----4

[tex]5xy+2yz=\lambda xyz[/tex] -----5

[tex]2yz+2xz=\lambda xyz[/tex]   ----6

Now equate 4 and 5

5xy+2xz=5xy+2yz

x=y

Substitute y=x in 5 and 6 and equate them

[tex]5x(x)+2(x)z=2(x)z+2xz\\5x^2=2xz\\5x=2z\\\frac{5}{2}x=z[/tex]

Substitute the values in A

[tex]V = xyz = x \times x \times \frac{5}{2}xV=\frac{5}{2}x^3\\\sqrt[3]{\frac{2}{5}V}=x\\x=y=\sqrt[3]{\frac{2}{5}V}\\z=\frac{5}{2}x=\frac{5}{2}(\sqrt[3]{\frac{2}{5}})=\sqrt[3]{\frac{25V}{4}}[/tex]

Hence,

The dimensions of the aquarium that minimize the cost of the materials:

[tex]x=y=\sqrt[3]{\frac{2V}{5}}\\z=\sqrt[3]{\frac{25V}{4}}[/tex]


Related Questions

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Answers

Answer:

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Step-by-step explanation:

To find the perpendicular slope, find the negative reciprocal, which is 4/13 in this case. All you do is flip the numerator and denominator, and change the sign.

Answer:

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Step-by-step explanation:

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A test contains two multiple-choice questions. If a student makes a random guess to answer each question. (Hint: Consider two outcomes for each question—either the answer is correct or it is wrong.) 2.2.1 How many outcomes are possible?

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Answers

Answer:

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Step-by-step explanation:

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Answers

Answer:

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The width of a rectangle is 50% more than the length. The perimeter is 65. Find both dimensions.

Answers

Answer:

Step-by-step explanation:

Perimeter of rectangle = 2 times width + 2 times length

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Answer:

Step-by-step explanation:

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Answers

Answer:

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Step-by-step explanation:

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Answers

Answer:

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Answers

Answer:

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Step-by-step explanation:

You don't want your credit to be low, and a higher number beats all of them.

You want high credit.

Not low.

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Answer:

Hey!

Here's your answer!!

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Answer:

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Step-by-step explanation:

Answer:

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Step-by-step explanation:

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Step-by-step explanation:

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Answers

Answer:

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Step-by-step explanation:

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Answers

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Answer:

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Step-by-step explanation:

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Answers

Answer:

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Step-by-step explanation:

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Answers

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Suppose you have information about average heights of a random sample of airline passengers. The mode is 65 inches, the median is 72 inches, and the mean is 70 inches. To convert the data into centimeters, multiply each data value by 2.54. What are the values of the mode, median, and mean in centimeters?

Answers

Answer: mode = 165.10 cm, median = 182.88 cm, mean = 177.80 cm

Step-by-step explanation:

mode = 65*2.54 = 165.10 cm

median = 72*2.54 = 182.88 cm

mean = 70*2.54 = 177.80 cm

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Mode = 162.56 cm

Median = 182.88 cm

Mean = 177.8 cm

Given,

The mode is 65 inches, the median is 72 inches, and the mean is 70 inches.

We need to convert the data into centimeters by multiplying each data value by 2.54 and find the values of the mode, median, and mean in centimeters.

How many cm in one inch?

There is 2.54 cm in one inch.

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If we want to find cm in 4 inches.

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2.54 cm = 1 inch

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We have,

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1 inch = 2.54 cm

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1 inch = 2.54 cm

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Median = 72 inches

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https://brainly.com/question/14881622

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Answers

9514 1404 393

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Step-by-step explanation:

Solving the equation for r gives ...

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Using the first line of the table, we find r to be ...

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DISTRICT SWI CA GRADE 6 MATH 20-21
D3101 2. P N →
Eat Test
4
BO
In a TV contest show, Thomas has 280 points. To determine his final score in the game, Thomas will spin the spinner shown below, and multiply his current number of points by the
fraction the spinner lands on
Which statement correctly describes the effect of multiplying his current score by the fractions shown on the spinner?
A Multiplying by will decrease his score
B Multiplying by will decrease his score.
Copyright 2020 laminate Education, Inc. All Rights Reserved
US 2:41

Answers

Answer:

Step-by-step explanation:

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Answers

Answer:

4/5 , 40/50, 80/100

Step-by-step explanation:

2) Martha cut three pieces of rope that are 5 1/4 inches each from a rope that is 20 inches
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Help me please

Answers

Answer:

Step-by-step explanation:

Total length of the rope = 20 inches.

Martha cuts 3 pieces of rope that is 5 1/4 inches each.

Find the total length of the 3 pieces of rope:

Total of 3 pieces of rope = 5 1/4 + 5 1/4 + 5 1/4 = 15.75

Since we know the total rope length is 20 inches, if you subtract the total length of the 3 pieces of rope you will get the remaining length of rope.

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Answers

Answer:

-4/5

Step-by-step explanation:

Answer:

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Step-by-step explanation:

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Answer:

Sunday: 50 Monday: 60 Tuesday: 50 Wednesday: 50 Friday: 40 Saturday: 40

Step-by-step explanation:

That's If Its To The Nearest Tenth If This Is What U Wanted I'm Just Doing What The Instructions Told Me

Ashley has sold 70% of the 20 candy bars she is supposed to sell for her softball team. How many candy bars does she have left to sell?

Answers

Answer:

70/100=x/20 cross multiply you get 1400/100=14

If Ashley sold 70% of her candy bars it means she has 30% left to sell

of means times

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Step-by-step explanation:

give brainliest please :)

Answer:

6

Step-by-step explanation:

70% of 20 is 14

20-14=6

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Enter the slope-intercept equation for the image of line ℓ after a clockwise rotation of 90°. (Hint: To find the image of line ℓ, choose two or more points on the line and find the images of the points.)​

Answers

Answer:

y = -2x - 2

Step-by-step explanation:

From the graph attached,

If the line 'l' passes through two points A(0, -1) and B(4, 1).

Rule for the rotation of any point (x, y), 90° clockwise about the origin is,

(x, y) → (y, -x)

By this rule,

A(0, -1) → A'(-1, 0)

B(4, 1) → B'(1, -4)

Let the equation of the line A'B' is,

y - y' = m(x - x')

where m = slope of the line A'B'

And (x', y') is a point lying on the line.

Slope of the line passing through A' and B' will be,

m = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]

m = [tex]\frac{0+4}{-1-1}[/tex]

m = -2

Therefore, equation of the line passing through (-1, 0) and slope = -2,

y - 0 = (-2)(x + 1)

y = -2x - 2

So the equation of the line after rotation of lie 'l' by 90° clockwise will be,

y = -2x - 2

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