Question

Congruent squares of length x are cut from the corners of a 10-inch by 15-inch piece of cardboard to create a box without a lid. Write an expression in terms of x for the length, the width, and the height of the box. Write a function, V(x), in terms of x for the volume of the box. Leave your answer in factored form. Express the reasonable domain of V(x) as an inequality and in interval notation.

Answer 1

To find the dimensions of the box created when congruent squares of length x are cut from the corners of a 10-inch by 15-inch piece of cardboard, we need to consider how these squares affect the length, width, and height of the box. The length of the box would be 15 - 2x, the width would be 10 - 2x, and the height would be x. The volume of the box, V(x), can be found by multiplying the length, width, and height: V(x) = (15 - 2x)(10 - 2x)(x).

Step-by-step explanation:

To find the dimensions of the box created when congruent squares of length x are cut from the corners of a 10-inch by 15-inch piece of cardboard, we need to consider how these squares affect the length, width, and height of the box.

When a square is cut from each corner, it reduces the length and width of the cardboard by 2x (since there are two corners on each side).

Therefore, the length of the box would be 15 - 2x, the width would be 10 - 2x, and the height would be x.

The volume of the box, V(x), can be found by multiplying the length, width, and height:

V(x) = (15 - 2x)(10 - 2x)(x)

The reasonable domain of V(x) can be expressed as an inequality: 0 < x < 5 (since the length and width cannot be negative or greater than the original dimensions).

In interval notation, the reasonable domain of V(x) is (0, 5).