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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

Y_Kistochka [10]

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

7 0

3 years ago

Is 2493 closer to 2400 rounded or 2,353

DiKsa [7]

2400 because it is only 93 away from 2493 and 2353 is more than 93 away

8 0

3 years ago

Read 2 more answers

A baseball player comes up to bat 3 times during a league game. He either gets a hit or gets an out. How many different combinat

Stella [2.4K]

Answer:

8 different combinations are possible.

Step-by-step explanation:

Here, we have 2 different combinations for each time.

And the player comes out to bat 3 times.

So, total number of combinations are:

$2^{3}$ i.e. a total of 8 number of times.

Let a hit is termed as 'H' and an out is termed as 'O'.

Total combinations are:

{HHH, HHO, HOH, HOO, OHH, OHO, OOH, OOO}

Kindly have a look at the tree diagram attached in the answer area.

In starting, there are 2 combinations possible, i.e. 'O' and 'H'.

After 'O' , 2 possible i.e. 'O' and 'H'.

After 'H' , 2 possible i.e. 'O' and 'H'.

and so on....

8 0

3 years ago

What is 3(x +3) -10 • 5

olga nikolaevna [1]

The answers is =3x−41

8 0

3 years ago

How many terms are in the algebraic expression 2y+3x-5x^2-g? 2 3 4 5

elena-14-01-66 [18.8K]

2y+3x-5x^2-g contains 4 terms that cannot be further simplified without know the values for the variables y, x, and g. Although 3x and -5x^2 contain the same variable x, since they have have different exponents, we can not directly add them together without knowing the value of x.

The answer is 4.

3 0

3 years ago