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.
let the side of the cut-out square be x inches
base is 10-2x by 10-2x, and the height is x
Volume = x(10-2x)^2 , where x > 5 , or else the base is negative.
To find the volume of the box, we need to first determine the dimensions of the base and the height. Let's start with the base.
When we cut squares of side length x from each corner of the square piece of cardboard, the resulting length of the base will be 10 - 2x (as we remove x from both ends). Similarly, the width of the base will also be 10 - 2x.
Now let's determine the height of the box. When we fold up the sides, the height is just the length of the squares we cut out, which is x.
The volume of the box can then be calculated by multiplying the length, width, and height:
Volume = (length) x (width) x (height)
= (10 - 2x) x (10 - 2x) x x
= x(10 - 2x)(10 - 2x)
So the volume of the box is given by the function V(x) = x(10 - 2x)(10 - 2x), where x represents the side length of the square cut out from each corner.