An open box is to be made from a rectangular piece of tin 12 inches long and 10 inches wide by cutting pieces of x-inch square from each corner and bending up the sides.
a. Find a formula that expresses the volume V of the box as a function of x.
b. Find the domain of the function.
v(x) = x(12-2x)(10-2x)
Since the sides must have positive length, we need 10-2x > 0, or x<5
Since the cuts must also have nonzero length, x>0, so the domain is
0 < x < 5
To find the formula for the volume of the box, we need to determine how the dimensions of the box change when we cut out x-inch squares from each corner.
a. Determining the dimensions of the box:
If we cut out x-inch squares from each corner, the resulting length of the base of the box will be (12 - 2x) inches, and the resulting width will be (10 - 2x) inches. The height of the box will be x inches.
b. Finding the volume:
The volume (V) of a rectangular box is given by the formula: V = length × width × height.
Substituting the values we found, we get:
V = (12 - 2x) × (10 - 2x) × x.
Simplifying this expression by multiplying out the terms gives us the formula for the volume of the box as a function of x:
V(x) = 4x^3 - 44x^2 + 120x.
Now, let's move on to finding the domain of the function.
The domain of a function refers to the set of possible inputs or values that can be plugged into the function. In this case, we need to consider the restrictions based on the dimensions of the rectangular piece of tin.
Since we are cutting squares from each corner, the length of the base of the box must be greater than 2x (to leave enough space for the sides to be bent up). Therefore, we have the inequality:
12 - 2x > 0.
Solving for x, we find:
-2x > -12,
x < 6.
This means the value of x must be less than 6 for the box to be physically possible and have positive dimensions.
Therefore, the domain of the function V(x) = 4x^3 - 44x^2 + 120x is x < 6. This indicates that x can take on any real number value less than 6.