Question

1. a box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. express the volume v of the box as a function of x.

Explanation:

Step1: Define box length

When we cut out squares of side $x$ from both ends of the 20 in side, the length of the box becomes $20 - 2x$.

Step2: Define box width

When we cut out squares of side $x$ from both ends of the 12 in side, the width of the box becomes $12 - 2x$.

Step3: Define box height

The height of the folded box is the side length of the cut squares, which is $x$.

Step4: Apply volume formula

The volume $V$ of a box is $\text{length} \times \text{width} \times \text{height}$. Substitute the expressions:
$$V(x) = x(20-2x)(12-2x)$$
We can expand this for clarity:
First multiply $(20-2x)(12-2x) = 240 - 40x - 24x + 4x^2 = 4x^2 - 64x + 240$
Then multiply by $x$:
$$V(x) = 4x^3 - 64x^2 + 240x$$
(Note: The domain is $0 < x < 6$ to keep all dimensions positive, but the function expression is the core result.)

Answer:

$V(x) = 4x^3 - 64x^2 + 240x$ (or factored form $V(x)=x(20-2x)(12-2x)$)