$\text { An open box of maximum volume is to be made from a square piece of material } 30$ centimeters on a side by cutting equal squares from the corners and turning up the sides(see figure). What is the domain of the function $V = x ( 30 - 2 x ) ^ { 2 }$.

A) domain: $0

Question

$\text { An open box of maximum volume is to be made from a square piece of material } 30$ centimeters on a side by cutting equal squares from the corners and turning up the sides(see figure). What is the domain of the function $V = x ( 30 - 2 x ) ^ { 2 }$.

A) domain: $0 < x < \infty$ 
B) domain: 30 
C) domain: $0 < x < 15$ 
D) domain: $0 < x < 30$ 
E) domain: 15

Quizplus · Accepted Answer

To find the domain of the function, we need to consider the restriction on the value of x, which is determined by the size of the original square piece of material and the size of the squares removed from each corner.

We can see from the figure that the height of the box is determined by the length of the square piece minus twice the length of the removed square:

height = 30 - 2x

The length and width of the base of the box are both equal to the side length of the original square minus twice the length of the removed square:

length = width = 30 - 2x

The volume of the box can be expressed as:

V = x(length)(width)(height)

Substituting in the expressions we found for length, width, and height:

V = x(30 - 2x)(30 - 2x)(30 - 2x)

Simplifying:

V = x(900 - 180x + 8x^2)

V = 8x^3 - 180x^2 + 900x

The domain of this function is restricted by the fact that the length of the squares removed from each corner must be shorter than half the side length of the original square piece of material, otherwise we won't be able to turn up the sides to form the box.

So we have the inequality:

x < 15

Since we also want the box to have positive volume, we need to exclude x = 0.

Therefore, the domain of the function is:

0 < x < 15

which corresponds to choice C.

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$\text { An open box of maximum volume is to be made from a square piece of material } 30$