Postal regulations specify that a package sent by certain shipping methods may have a combined length and girth of no more than 99 inches. Find the dimensions of a CYLINDRICAL package with the greatest volume that may be sent by this shipping method. HINT* the length plus the girth is 2pi(radius)+ length 0r 2pir+l.
99=L+2PI*radius
volume=PI*r^2 * L=PI r^2*(99-2PIr)
dv/dr=2PI r (99-2PIr)+PIr^2 *-2pi)
set dv/dr=0
and solve for r.
To find the dimensions of the cylindrical package with the greatest volume that can be sent, we need to determine the values of the radius and length that satisfy the given constraint.
Let's start by understanding the formula provided in the hint: length plus the girth is equal to 2πr + length (2πr + l).
Since the postal regulations specify that the combined length and girth cannot exceed 99 inches, we can set up the equation:
2πr + l ≤ 99
Now, let's express the volume of a cylinder using the formula V = πr²h, where "V" represents volume, "r" is the radius, and "h" is the height/length.
To maximize the volume, we need to maximize both the radius and the length, while still satisfying the given constraint.
Since the combined length and girth is 2πr + l, we can rearrange the equation to solve for l:
l = 2πr + l - 2πr
Simplifying, we get:
l = 2πr + l - 2πr
l = l
This equation tells us that the length could be any value without any impact on the constraint. Thus, we can choose a value of l that simplifies our calculations.
Let's choose l = 0, which means the package has no length or height. In this case, the formula for the combined length and girth becomes:
2πr + 0 = 2πr
Now, we can rewrite the constraint equation with l = 0:
2πr ≤ 99
Dividing both sides of the inequality by 2π, we have:
r ≤ 99 / (2π)
With this value for r, the radius, and l = 0, the length, the cylindrical package will have its maximum volume while still satisfying the given constraint.
To find the exact dimensions, you can substitute the value of r into the equation for volume: V = πr²h. But since the length is 0, the volume will also be 0.
Therefore, the cylindrical package with the greatest volume that can be sent by this shipping method has a radius of (99 / (2π)) and a length of 0.