The USPS will not mail a package if the sum of its length and its girth (measuring around the package) exceed 108 inches. Determine:
(a) The volume of the package as a function of its radius r.
(b) For which radius r is the volume of the package a maximum?
(c) What is the maximum volume of the package?
h = length
r = radius of cylinder package (I assume)
girth = 2 pi r
h + 2pi r = 108 for maximum
so h = (108-2pir)
V = pi r^2 h
V = pi r^2 (108-2 pi r)
dh/dr = pi r^2 (-2pi) + (108-2 pi r)2pi r
for max
2 pi r^2 = 216r - 4 pi r^2
6 pi r^2 = 216 r
r = 216/(6 pi)
r = 11.5
then h = 108 -2 pi r = 36
then V = pi r^2 h = 14851 in^3
radius? Radius is on spheres. What kind of package are you askig about.
The question never specifies the shape of the package so it has really confused me on how to solve it.
Well by giving you r, it kind of implies a right circular cylinder.
To determine the volume of the package as a function of its radius, we first need to understand the formula for calculating the volume of a cylinder.
The volume of a cylinder is given by the formula:
V = πr^2h
In this case, the length and girth of the package are mentioned. The girth is the distance around the package, which is 2πr (since it is a cylinder).
The USPS will not mail a package if the sum of its length and girth exceed 108 inches. So we can write the equation as:
L + 2πr > 108
Now, let's solve for h (height) in terms of r:
L + 2πr = 108
L = 108 - 2πr
h = 108 - 2πr
To determine the volume of the package as a function of its radius r, substitute the value of h in the volume formula:
V = πr^2h
V = πr^2(108 - 2πr)
(a) The volume of the package as a function of its radius r is V = πr^2(108 - 2πr).
To find the radius r for which the volume of the package is a maximum, we can use calculus. We need to find the value of r that makes the derivative of the volume function equal to zero.
(b) Let's differentiate the volume function with respect to r and set it equal to zero:
dV/dr = 2πr(108 - 2πr) + πr^2(-2π) = 0
Simplifying the equation:
2πr(108 - 2πr) - 2π^2r^2 = 0
r(108 - 2πr - 2πr) = 0
r(108 - 4πr) = 0
Setting each factor equal to zero:
r = 0 (Not feasible since the radius cannot be zero)
108 - 4πr = 0
Solving for r:
108 = 4πr
r = 108/(4π)
r = 27/π
(c) To find the maximum volume of the package, substitute the value of r back into the volume equation:
V = πr^2(108 - 2πr)
V = π(27/π)^2(108 - 2π(27/π))
V = 27^2(108 - 54)
V = 27^2(54)
V = 27^2 * 54
V = 19683 * 54
V = 1,061,802 cubic inches
Therefore, the maximum volume of the package is 1,061,802 cubic inches when the radius is 27/π inches.