Postal regulations specify that a package sent by priority mail may have a combined length and girth of no more than 129 in. Find the dimensions of a cylindrical package with the greatest volume that may be sent by priority mail. Hint: The length plus the girth is 2πr + l.
To find the dimensions of a cylindrical package with the greatest volume that may be sent by priority mail, we'll need to use some calculus principles, specifically optimization.
Let's denote the radius of the cylindrical package as "r", and the length of the package as "l". The formula for the combined length and girth is given as: L + G = 2πr + l, where L is the length, and G is the girth.
According to the postal regulations, L + G must be less than or equal to 129 inches.
To find the dimensions with the greatest volume, we'll maximize the volume function V(r, l) = πr^2l, which represents the volume of the cylinder.
Now, we have two conditions:
1. L + G ≤ 129
2. We want to maximize V(r, l) = πr^2l
To begin, let's solve condition 1 for l:
l = 129 - 2πr
Substituting this value for l in the volume function, we get:
V(r) = πr^2(129 - 2πr)
Expanding and simplifying, we get:
V(r) = 129πr^2 - 2π^2r^3
Now, we'll find the critical points by taking the derivative of V(r) with respect to r and setting it equal to zero:
V'(r) = 258πr - 6π^2r^2 = 0
Simplifying further, we have:
3r(86 - πr) = 0
Setting each term equal to zero and solving for r, we find two possible critical points:
r = 0 or r = 86/π
Since r = 0 would give a cylinder of zero volume, we'll discard this value.
Now, let's find the value of l for the critical point r = 86/π using the equation we derived earlier:
l = 129 - 2πr
l = 129 - 2π(86/π)
l = 129 - 172
l = -43 (which is not a valid length, as lengths must be positive)
Therefore, we can conclude that the critical point r = 86/π corresponds to an invalid solution, and there isn't a maximum volume that satisfies the postal regulations.
In this case, it means there is no cylindrical package with the greatest volume that can be sent by priority mail within the given postal regulations.