A parcel delivery service will deliver a package only if the length plus girth (distance around) does not exceed 84 in. What are the radius, length and volume of the largest cylindrical package that may be sent using this service?
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To find the largest cylindrical package that can be sent using the delivery service, we need to determine the maximum length, radius, and volume of the cylinder while ensuring that the length plus girth (distance around) does not exceed 84 inches.
Let's start by understanding the dimensions of a cylindrical package. A cylinder has two circular bases with a radius (r) and a height (h). The length plus girth is essentially the distance around the cylinder, which is the sum of the length and the circumference of the circular base.
So, the formula for the length plus girth of a cylinder is:
Length + Girth = Length + 2πr
Since the length plus girth must not exceed 84 inches, we have the following inequality:
Length + 2πr ≤ 84
Next, let's consider the volume of the cylinder. The volume (V) of a cylinder is given by the formula:
Volume = πr^2h
To find the largest volume, we need to maximize both the radius and height of the cylinder, while satisfying the length plus girth constraint.
To visually understand the problem, let's assume the length (L) is equal to the height (h) of the cylinder. Now, we can rewrite the constraint as:
L + 2πr ≤ 84
Substituting L for h, we get:
h + 2πr ≤ 84
To maximize the volume, we need to maximize both r and h. Since the value of h is equal to L, we can rewrite the inequality as:
L + 2πr ≤ 84
L + 2πr ≤ 84
To find the maximum volume, we can use calculus to find the optimal values of r, L, and h. However, since this is a simple problem, we can solve it algebraically.
First, let's rearrange the inequality to solve for L:
L ≤ 84 - 2πr
Since L = h, we have:
h ≤ 84 - 2πr
To make the volume as large as possible, we need to consider the largest radius that satisfies the constraint. Now, let's find the maximum value of r by taking the derivative of the inequality with respect to r and setting it equal to zero:
d(h)/dr = 0
We differentiate the equation and set it equal to zero:
d/dx (84 - 2πr) = 0
-2π = 0
Since -2π is a constant value, it cannot be equal to zero. Therefore, there is no maximum value for r.
This means that there is no limit on the radius (r) of the cylinder. However, the maximum length (L) can be determined by substituting the maximum value of r into the inequality. Let's do that:
L + 2π * 0 ≤ 84
Simplifying, we get:
L ≤ 84
Therefore, the maximum length of the cylinder is 84 inches.
As for the volume, we can substitute the maximum values of L and r into the volume formula:
Volume = πr^2h
Volume = π * (0)^2 * 84
Volume = 0
Hence, the maximum volume of the largest cylindrical package that may be sent using this service is 0 cubic inches. This means that the package has no dimensions and cannot exist.