UBC parcel post regulations states that packages must have length plus girth of no more than 84 inches. Find the dimension of the cylindrical package of greatest volume that is mailable by parcel post. What is the greatest volume? Make a sketch to indicate your variables.
I have no idea where to start with this question. Please show work so I can follow along :)
Thanks in advance!
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A parcel service will only accept packages with a length of no more than 96 inches and length plus girth of no more than 156 inches. (See the figure below.) Assuming that the front face of the package (as shown in the figure) is square, what is the largest volume package that the parcel service will accept?
31246
To find the dimensions of the cylindrical package of greatest volume that is mailable by parcel post, we need to maximize the volume of a cylinder subject to the constraint that the length plus girth does not exceed 84 inches.
Let's start by defining some variables. Let:
- r be the radius of the cylinder,
- h be the height of the cylinder, and
- V be the volume of the cylinder.
We need to find the values of r, h, and V that maximize the volume while satisfying the constraint: length plus girth <= 84 inches.
The length of the cylinder is equal to the height (h), and the girth is equal to the circumference of the circular base of the cylinder, which is 2πr.
Therefore, the constraint can be written as:
h + 2πr <= 84 inches.
To solve for the dimensions of the cylinder of maximum volume, we need to express the volume V in terms of a single variable. Since the volume of a cylinder is given by V = πr^2h, we can substitute the constraint equation to eliminate h:
V = πr^2(h) = πr^2(84 - 2πr).
Now, we have the volume V as a function of r only. To find the maximum volume, we need to find the critical points by differentiating V with respect to r and setting it equal to zero:
dV/dr = 2πr(84 - 2πr) + πr^2(-4π) = 2πr(84 - 6πr) = 0.
Solving this equation for r, we get two possible values: r = 0 and r = 14/π.
Since r cannot be zero (otherwise, the cylinder does not exist), the only valid value is r = 14/π.
Now that we have the value of r, we can substitute it back into the constraint equation to solve for h:
h + 2πr = 84,
h + 2π(14/π) = 84,
h + 28 = 84,
h = 84 - 28,
h = 56 inches.
Therefore, the height of the cylinder is 56 inches, and the radius is 14/π inches.
Finally, we can find the maximum volume by substituting the values of r and h into the volume equation:
V = π(14/π)^2(56) = 2744 cubic inches.
So, the greatest volume of the cylinder that is mailable by parcel post is 2744 cubic inches.
To summarize the dimensions of the cylinder:
- Radius (r) = 14/π inches,
- Height (h) = 56 inches.
I hope this explanation helps you understand how to solve the problem.
Let
L=length, then
girth=84-L=2*radius=2R, or
R=(84-L)/2
Volume, V = πR²L
Express V in terms of L using R=(84-L)/2
Equate dV/dL=0 and solve for L.