Question

The terrarium is in the shape of a rectangular prism. The volume of the terrarium is 4000 cubic inches. The length is greater than the height. What are the dimensions of the terrarium? A rectangular prism with a height labeled x minus 4 inches. A width of 10 inches. A length of x plus 5 inches.

Answer 1

length = 25 inches

height = 16 inches

Explanation:

Volume of rectangular prism:

width = w = 10 inches

height = h = (x - 4) inches

length = l = (x + 5) inches

`\boxed{\text{\bf Volume of rectangular prism = l * w * h}}`

l *w * h = 4000 cubic inches

Substitute the values of l, w and h in the above equation.

(x + 5) * 10 * (x - 4) = 4000

Divide the entire equation by 10,

(x + 5) (x - 4) = 4000 ÷ 10

(x + 5) (x - 4) = 400

Use FOIL method to multiply the expressions in LHS,

x*x - x*4 + 5*x - 5 *4 = 400

x² - 4x + 5x - 20 = 400

Combine like terms,

x² + x - 20 = 400

x² + x - 20 - 400 = 0

x² + x - 420 = 0

Sum = 1

Product = -420

Factors = 21, (-20)

When we add 21 + (-20), it gives 1 and when we multiply 21 & (-20), it gives -420.

Rewrite the middle terms using the factors,

x² + 21x - 20x - 420 = 0

x (x + 21) - 20(x + 21) = 0

(x + 21) (x - 20) = 0

x + 21 = 0 or x -20 = 0

x = -21 x = 20

x = -21 is rejected as dimensions cannot be a negative number.

So, x = 20 inches

length = x + 5

= 20 + 5

= 25 inches

height = x - 4

= 20 - 4

= 16 inches

Answer: length = 25 inches

height = 16 inches