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

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

The values in cm are:

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.

Example:

If we want to find cm in 4 inches.

We just multiply 4 with 2.54 cm.

2.54 cm = 1 inch

4 x 2.54 cm = 4 x 1 inch

10.16 cm = 4 inches

We have,

We know that,

1 inch = 2.54 cm

Mode = 65 inches

1 inch = 2.54 cm

64 inches = 64 x 2.54 cm = 162.56 cm

Median = 72 inches

72 inches = 72 x 2.54 cm = 182.88 cm

Mean = 70 inches

70 inches = 70 x 2.54 cm = 177.8 cm

Thus the values in cm are:

Mode = 162.56 cm

Median = 182.88 cm

Mean = 177.8 cm

Learn more about the conversion of inches to cm here:

https://brainly.com/question/14881622

#SPJ2

Need help ASAP !!! I am failing MATH

Answers

Answer:

i think it is D

Step-by-step explanation:

When do we use inequalities?

Answers

Answer: In math but if u mean “irl” they use it in business like how much is in there inventory (stock) and the price stock they can make to get an profit and etc.

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
long. What is the length of the remaining part of the rope?
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.

Remaining length = 20 - 15.75

= 4.25 inches of rope

The Madison's son Dwayne

Answers

What does this mean? lol

Given a linear equation with a slope of - 13/4, determine the slope of a line that is perpendicular. Enter the answer as a fraction.

Answers

Answer:

4/13

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:

4/13 should be the answer as it is the reciprocal.

Step-by-step explanation:

I learned this like 2 or 3 years ago so im a bit rusty but im pretty sure thats the answer

Do you wants to buy a bike for $180 he has $40 Dave mows lawns for $10 per lawn Write in equation that can be used to find eggs the number of lawns Dave needs mow to have enough money to the to buy his bike

Answers

Answer:

Dave needs to mow 14 lawns to have enough money to buy a bike.

Step-by-step explanation:

180-40=140, 140/10=14. Dave needs to mow 14 lawns to have enough to buy a bike.

Dennis mowed his next door neighbors lawn for a handful of dimes and nickels, 80 coins in all. Upon completing the job he counted out the coins and it came to $6.60. How many of each coin did he earn

Answers

Answer:

28 nickels and 52 dimes

Step-by-step explanation:

.05x 28 + .10 x 52 = 6.60

1.40 + 5.20 = 6.60

^28  +   ^52  = ^80

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

this is due today help

Answers

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

Monica works in a mailroom for 18 hours
each week. How many hours does she work in
52 weeks?

Answers

Answer:

936

Step-by-step explanation:

Because 18 times 52= 936

student schoolitycom/students/common/index
Cow
CCM! Payne New
Henry, Ladaris
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:

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

30%=.30

20 x .30= 6 candy bars to sell

Step-by-step explanation:

give brainliest please :)

Answer:

6

Step-by-step explanation:

70% of 20 is 14

20-14=6

let me  know if im wrong

What did Physics say to History when they are arguing?

Answers

Answer:

Hey!

Here's your answer!!

➡️Don't change the subject⬅️

Why do they even exist!?

Gotta study those experiments which we never perform and those people who are long gone xD

find three ratios that are equivalent to 20:25

Answers

Answer:

4/5 , 40/50, 80/100

Step-by-step explanation:

4(a + 2) + 2 = 4 - 2(-2a - 3)

Answers

Answer:

0=0

Step-by-step explanation:

4(a + 2) + 2 = 4 - 2(-2a - 3)

4a +  8 + 2 = 4 + 4a + 6

4a + 10 = 10 + 4a

0 = 0

A family buys 8 tickets to a show. They also pay a $5
parking fee. They spend $61 to see the show.
What is the solution to the story?

13.80
10.60
07.00
8.25

Answers

Answer:

C. $7.00

Step-by-step explanation:

8 times 7 is 56 + 5 = 61

What is the slope of the line that passes through the points (1, 1) and (7, 5)?

Answers

Answer:

The slope of the line is

[tex] \frac{2}{3} [/tex]

What is the sum of the fractions? Use the number line to help find the answer.

3/5 plus -7/5

Answers

Answer:

-4/5

Step-by-step explanation:

Answer:

The answer is -4/5

Step-by-step explanation:

Hope this helps!:)

used to parallel lines cut by transversal below to answer the question. which statement is not a true statement will mark brainest ​

Answers

Answer:

B

Step-by-step explanation:

there is no angle e unless i am reading this incorrectly, but i double-checked all the other answer choices and they are all correct, so B is the answer:)

Answer:

see attached

Step-by-step explanation:

If Coach Smith has 12 players and each player needs three uniforms, three shin guards, two balls, and three pairs of cleats. How much will he end up spending on his equipment for the entire soccer team?

Answers

you can't do that. there isn't enough detail. we don't know how much each item costs

HELP ASAPPPP
What is the equation of this graphed line? If the line passes through (-6,-3) and (6,-7)

Answers

Answer:

y= -1/3x - 5

Step-by-step explanation:

Answer:

The answer would be y=-1/3x-7.

Step-by-step explanation:

Using the formula y2-y1/x2-x1 to find the slope of the line, plug in the numbers and you will get a slope of -1/3. Now using the the formula y-y1=m(x-x1) and plug in the numbers, and you will get the equation y=-1/3x-7.

A dolphin jumped from 5 feet below the surface of the sea. Jane wrote −5 to represent the depth from which the dolphin jumped. What does 0 represent in this situation?

Answers

0 represent the sea surface since the dolphin jumped from five feet below the sea surface and since Jane put that the dolphin jumped at -5 below the sea surface that would mean that 0 must represent the sea surface.

Answer:

it represents the surface of the sea

Step-by-step explanation:

i did the test lol :)

Alice is completing the table so that it is a function. Which values can she NOT use to complete
the table? Explain.

Answers

It’s the 2nd option

A restaurant manager classifies customers as regular, occasional, or new

Answers

Answer:

Step-by-step explanation:

I am sorry but please give detailed question

The quantities and are proportional​

Answers

9514 1404 393

Answer:

  r = 1

Step-by-step explanation:

Solving the equation for r gives ...

  r = y/x

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

  r = 5.8/5.8 = 1

The constant of proportionality is r = 1.

What is a better credit score?
Select the correct answer below:
A lower number
An even number
An odd number
O A higher number​

Answers

Answer:

A higher number.

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.

Answer:

definately you want a higher number

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

P= 2W + 2L

Let's say W = width , L= length

We are told the width is 50% more than the length.

width = 0.5 + L

W = 0.5 + L

From here we can say

Perimeter = 2W + 2L so,

P = 2(0.5 + L) + 2L

P = 1 + 2L + 2L

P = 1 + 4L

We are also told perimeter is 65 (P=65)

65 = 1 + 4L

Subtract 1 from both sides

64 = 4L

Divide 64 by 4 to find L

L = 64/4 = 16

Now since we know L= 16, this means

W = 0.5 + L is the same as

W = 0.5 + 16 = 16.5

Therefore Length = 16, width = 16.5

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?

Answers

Answer: 2

Step-by-step explanation: he could ethier get t right or get it wrong

can you help me on this question
they are the same but one of the is a bad pic

Answers

0.000017 in scientific notation would be: 1.7 x 10^-5
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