A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 90 inches (see figure). Find the dimensions of the package of maximum volume that can be sent. (Assume the cross section is square.)
To find the dimensions of the package of maximum volume that can be sent, we need to maximize the volume of a rectangular package under the given constraint.
Let's assume the length of the rectangular package is x inches. Since the cross-section is a square, the width and height of the package will also be x inches.
The perimeter of the cross-section (girth) is the sum of the four sides of the square, which is 4x inches.
According to the given constraint, the combined length and girth of the package should be less than or equal to 90 inches. So, we can write the following equation:
Length + Girth ≤ 90
x + 4x ≤ 90
5x ≤ 90
Dividing both sides of the equation by 5, we have:
x ≤ 18
Now, let's find the volume of the rectangular package in terms of x:
Volume = Length × Width × Height
= x × x × x
= x^3
To maximize the volume, we need to find the maximum value of x^3.
Since x ≤ 18 (from the previous equation), we can test different values of x starting from 1 to find the value that gives the maximum x^3. But we can also take the derivative of x^3 with respect to x and set it equal to 0 to find the critical points.
Let's find the derivative:
d(x^3)/dx = 3x^2
Setting the derivative equal to 0:
3x^2 = 0
x^2 = 0
x = 0
We have found a critical point at x = 0. However, this is not a valid solution since the length, width, and height of the package cannot be zero.
So, we conclude that the maximum value of x that satisfies the constraint is x = 18 inches.
Therefore, the dimensions of the package of maximum volume that can be sent are:
Length = 18 inches
Width = 18 inches
Height = 18 inches
Let's denote the length of the rectangular package as L and the width as W. Since the cross-sectional shape is square, we can assume that the length and width are equal, so L = W.
The girth of the package is the perimeter of the cross-section, which is 2L + 2W. According to the problem, the maximum combined length and girth is 90 inches. So we can write the equation:
L + girth = 90
Substituting the value of girth with 2L + 2W, we get:
L + 2L + 2W = 90
Combining like terms, we have:
3L + 2W = 90
Since L = W, we can substitute L for W in the equation:
3L + 2L = 90
Simplifying, we get:
5L = 90
Dividing both sides of the equation by 5, we find:
L = 18
Therefore, the length of the package is 18 inches. Since L = W, the width of the package is also 18 inches.
To find the height of the package, we need to consider the maximum volume. The volume of a rectangular package is given by V = L x W x H. In this case, L = W = 18 inches, so the volume becomes:
V = 18 x 18 x H
To maximize the volume, we need to maximize H. The package has a maximum combined length and girth, so it is essentially a cylinder. The height of a cylinder is its longest dimension, which in this case is 18 inches.
Therefore, the dimensions of the package of maximum volume that can be sent are: Length = 18 inches, Width = 18 inches, and Height = 18 inches.