A rectangular box with no top is to be constructed from a 10 in. x 10 in square piece of cardboard by cutting equal square of side x from each corner and then bending up the sides. Write the volume of the box as a function of x.
To write the volume of the box as a function of x, we need to determine the dimensions of the box in terms of x and then use those dimensions to calculate the volume.
Let's start by considering the original square piece of cardboard. Since equal squares are cut from each corner, the remaining piece of the cardboard will have sides of length 10 - 2x (since x is the side length of the square cut from each corner).
When we fold up the sides to form the box, the height of the box will be equal to x.
Therefore, the dimensions of the box will be:
Length: 10 - 2x
Width: 10 - 2x
Height: x
Now, we can calculate the volume by multiplying the dimensions:
Volume = Length * Width * Height
= (10 - 2x) * (10 - 2x) * x
= (10 - 2x)^2 * x
= (100 - 40x + 4x^2) * x
= 4x^3 - 40x^2 + 100x
So, the volume of the box is given by the function V(x) = 4x^3 - 40x^2 + 100x.