A manufacturer of open tin boxes wishes to make use of pieces of tin with dimensions 8 in. by 15 in. by cutting equal squares from the four corners and turning up the sides.
a. Let x inches be the length of the side pf the square to be cut out; express the number of cubic inches in the volume of the box as a function of x.
b. what is the domain of the resulting function?
a. v = x (8 - 2x) (15 - 2x)
b. x must be less than 4 (the point where the 2nd factor goes to zero)
B.What is the domain of the resulting function
To find the volume of the box, we need to subtract the volume of the square cutouts from the original dimensions.
a. Let's start by visualizing the original box before any cuts are made. The length, width, and height of the box are given as 15in, 8in, and x inches respectively.
After cutting out squares with sides of length x from each corner, the resulting dimensions of the box will be:
Length: 15in - 2x
Width: 8in - 2x
Height: x
The volume of a box is calculated by multiplying its length, width, and height.
So, the volume V(x) in cubic inches of the box can be expressed as:
V(x) = (15in - 2x) * (8in - 2x) * x
b. Now, let's determine the domain of the resulting function.
The domain represents the possible values of x for which the function is defined. In this case, we need to consider the restrictions based on the dimensions of the original tin piece and the practicality of cutting out the squares.
1. The side length of the square, x, cannot be greater than half the length or width of the original piece, as it would result in negative dimensions for the box. So, we have the following inequality:
x <= min(15in/2, 8in/2) = min(7.5in, 4in) = 4in
2. The side length of the square, x, also cannot be greater than the height of the original piece, as it would require cutting more than the available material. So, we have:
x <= x
Considering both conditions, the domain of the function is:
0 <= x <= 4in
Therefore, the resulting function is defined for values of x between 0 and 4 inches (inclusive).