Story problem help please

Question

Please help me solve this. Thanks!

An aquarium of height 1.5 feet is to have a volume of 6 ft^3. Let x denote the length of the base and y the width. (Assume the aquarium has an open top.)

 

(a) Express y as a function of x.
y(x) =



(b) Express the total number S of square feet of glass needed as a function of x.
S(x) =

Answer 1

Since we have a length, width, and height, we have a rectangular prism.

 

a)

 

The volume of a rectangular prism is the product of those dimensions.

 

xyh = V

 

 

Plug in the know values into this equation.

 

xy(1.5) = 6

 

xy(3/2) = 6

 

xy = 6(2/3)

 

xy = 4

 

  y = 4 / x

 

 y(x) = 4 / x

 

 

b)

 

We take the surface area of the prism.  Keep in mind of the open top.  However, the glass is only applied to the sides.

 

SA = 2xh + 2yh

 

 

Then plug in the value of y from the previous solution as well as h so that the total area is in terms of x.

 

S(x) = 2x(3/2) + 2(4/x)(3/2)

 

S(x) = 3x + 12/x 

 

S(x) = (3x² + 12) / x