1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
san4es73 [151]
3 years ago
13

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(Aquarium has rectangular base).
Mathematics
1 answer:
Y_Kistochka [10]3 years ago
7 0

Answer:

x = ∛ 2*V/5  

y = ∛ 2*V/5

h  = V/ ∛ 4*V²/25

Step-by-step explanation:

Dimensions of the aquarium base is  x*y

We call c₁ cost per unit area of the sides, then cost per unit area of slate is equal 5c₁.

let call h the height of the aquarium then volume of the aquarium is:

V = x*y*h      where   h =  V / x*y

As the base is a rectangular one there are 2 sides x*h .  and 2 sides  y*h

According to this:

Ct (cost of aquarium )  = cost of the base  + cost of the sides

cₐ  ( cost of the base) = 5*c₁*x*y

c₆ (cost of the sides ) = c₁*2*x*h   +   c₁*2*y*h

C(t)  =  5*c₁*x*y +2* c₁*x* V/x*y  +  2* c₁*y* V/x*y    or

C(t)  =  5*c₁*x*y  + 2*c₁*V/y   *2*c₁* V/x

Taking partial derivatives en x and y we have:

C´(x)  =  5*c₁*y - 2*c₁*V/x²

C´(y)  =  5*c₁*x - 2*c₁*V/y²

C´(x)  = C´(y)        ⇒  5*c₁*y - 2*c₁*V/x²  =   5*c₁*x - 2*c₁*V/y²

or    5*y - 2*V/x²  =   5*x - 2*V/y²

(5*y*x² - 2*V)/x²  = ( 5*y²x - 2*V) /y²

(5*y*x² - 2*V)*y²  = ( 5*y²x - 2*V)*x²

5*y³*x² - 2*V*y²  =  5*y²x³  - 2*V*x²

5*y³*x² - 5*y²x³  =  2*V * ( y² - x²)

by symmetry  x =  y

Then using x = y  and plugging that value on the derivatives

C´(x) =  5*c₁*y - 2*c₁*V/x²

C´(x) =  5*c₁*x - 2*c₁*V/x²

C´(x) = 0          ⇒     5*c₁*x - 2*c₁*V/x²  = 0

5*x  - 2*V/x² = 0      ⇒  5*x³ - 2*V = 0   ⇒   5*x³  = 2*V  ⇒ x³ = 2*V/5

x = ∛ 2*V/5       and   y = ∛ 2*V/5    and   h  =  V/ x*y    h  = V/ ∛ 4*V²/25

You might be interested in
Please help with these 2 answers i only have 10 minutes left
natta225 [31]

Answer:

  1. 1/4 of a pound
  2. 4/15 of the flat

Step-by-step explanation:

For number 1:

  1. 3/4 ÷ 3 = 0.25
  2. 0.25 = 1/4

For number 2:

  1. 4/5 ÷ 3 = 4/15

I hope this helps!

5 0
3 years ago
Read 2 more answers
Which statements are true?
Alexxx [7]
B. All rectangles are parallelograms! And C. All equilateral triangles are isosceles! :-)
7 0
3 years ago
Read 2 more answers
Find the missing angles in the question blow 22 inches a
kondaur [170]
Where the rest of the question?
8 0
2 years ago
Plz help me with this plz!!!!!
Kipish [7]
It’s bull Peter C your welcome :)
7 0
3 years ago
3 times a number m use a equation plz
lawyer [7]

Answer: 3m

Step-by-step explanation:

You're multiply 3 by m. Since this isn't possible unless you have m's value, you must write the expression as 3m.

7 0
3 years ago
Other questions:
  • a company is offering a job with a salry of 30000 for the first year and a 5% raise each year after that If the 5% raise continu
    14·1 answer
  • Darla spent $120.37 on groceries in January, $108.45 in February, and $114.86 in March. What was the average monthly cost of Dar
    9·2 answers
  • 3806 divided by 22=?
    12·2 answers
  • Molly shot 20 baskets in 4 minutes nicko's shot 42 baskets and 6 minutes how many more baskets did Nico shoot per minute
    6·2 answers
  • Graph the following solution y=-1/2×+1 and y=-2/3×+2
    6·1 answer
  • A number is divided by 2, and then 8 is subtracted from the quotient. The result is 16. What is the number?
    7·2 answers
  • 30 POINTS!!!! and i will give bairnless plzz
    12·2 answers
  • What is <br> 1/2 times 1/8
    6·1 answer
  • The distance from Jamestown to Kenneville is 180 miles, the distance from Kenneville to Elizabeth City is 300 miles, and the dis
    8·1 answer
  • Question 3
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!