To find the dimensions of a box that will give the maximum volume under the given conditions, let's go through the problem step by step.
1. Set up the problem:
- We know the perimeter of the top of the box is 72 inches.
- The box is 8 inches high.
2. Express the perimeter equation:
- Let's denote the length of the rectangle as [tex]\( l \)[/tex] and the width as [tex]\( w \)[/tex].
- The perimeter of a rectangle is given by:
[tex]\[
2l + 2w = 72
\][/tex]
- Simplify this equation:
[tex]\[
l + w = 36
\][/tex]
Therefore, we can express [tex]\( w \)[/tex] in terms of [tex]\( l \)[/tex]:
[tex]\[
w = 36 - l
\][/tex]
3. Express the volume equation:
- The volume of the box [tex]\( V \)[/tex] is given by the product of the length, width, and height:
[tex]\[
V = l \cdot w \cdot h
\][/tex]
- We know the height [tex]\( h \)[/tex] is 8 inches, so:
[tex]\[
V = l \cdot (36 - l) \cdot 8
\][/tex]
Simplify this to:
[tex]\[
V = 8l (36 - l)
\][/tex]
Which further simplifies to:
[tex]\[
V = 8(36l - l^2)
\][/tex]
4. Find the maximum volume by optimizing the quadratic expression:
- This is a quadratic function of the form [tex]\( V = 8(36l - l^2) \)[/tex], which can be rewritten as:
[tex]\[
V = -8l^2 + 288l
\][/tex]
- The quadratic function [tex]\( V = -8l^2 + 288l \)[/tex] opens downward (as the coefficient of [tex]\( l^2 \)[/tex] is negative), and its maximum value occurs at its vertex.
- The vertex form of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] has its maximum (or minimum) at:
[tex]\[
l = -\frac{b}{2a}
\][/tex]
- Here, [tex]\( a = -8 \)[/tex] and [tex]\( b = 288 \)[/tex]:
[tex]\[
l = \frac{-288}{2 \cdot -8} = \frac{288}{16} = 18
\][/tex]
5. Calculate the corresponding width:
- Substitute [tex]\( l = 18 \)[/tex] back into the width equation:
[tex]\[
w = 36 - 18 = 18
\][/tex]
6. Calculate the maximum volume:
- Substitute [tex]\( l = 18 \)[/tex], [tex]\( w = 18 \)[/tex], and [tex]\( h = 8 \)[/tex] into the volume equation:
[tex]\[
V = 18 \cdot 18 \cdot 8 = 2592
\][/tex]
So, the dimensions that give the maximum volume are a length of 18 inches, a width of 18 inches, and the height as given, 8 inches. The maximum volume of the box is 2592 cubic inches.
