Respuesta :
Answer:
The dimensions of the package is [tex]r=\frac{48}{\pi}\ \text{and} \ h=48[/tex].
Step-by-step explanation:
Consider the provided information.
As it is given that, cylindrical package to be sent by a postal service can have a maximum combined length and girth is 144 inches.
Therefore,
144 = 2[tex]\pi[/tex]r + h
144-2[tex]\pi[/tex]r = h
The volume of a cylindrical package can be calculated as:
[tex]V=\pi r^{2}h[/tex]
Substitute the value of h in the above equation.
[tex]V=\pi r^{2}(144-2\pi r)[/tex]
Differentiate the above equation with respect to r.
[tex]\frac{dV}{dr}=2\pi r(144-2\pi r)+\pi r^{2}(-2\pi)[/tex]
[tex]\frac{dV}{dr}=288\pi r-4{\pi}^2 r^{2}-2{\pi}^2 r^{2}[/tex]
[tex]\frac{dV}{dr}=288\pi r-6{\pi}^2 r^{2}[/tex]
[tex]\frac{dV}{dr}=-6\pi r(-48+\pi r)[/tex]
Substitute [tex]\frac{dV}{dr}=0[/tex] in above equation.
[tex]0=-6\pi r(-48+\pi r)[/tex]
Therefore,
[tex]0=-48+\pi r[/tex]
[tex]r=\frac{48}{\pi}[/tex]
Now, substitute the value of r in 144-2[tex]\pi[/tex]r = h.
[tex]144-2\pi\frac{48}{\pi}=h[/tex]
[tex]144-96=h[/tex]
[tex]48=h[/tex]
Therefore the dimensions of the package should be:
[tex]r=\frac{48}{\pi}\ \text{and} \ h=48[/tex]
This is about optimization problems in mathematics.
Dimensions; Height = 48 inches; Radius = 48/π inches
- We are told the combined length and girth is 144 inches.
Girth is same as perimeter which is circumference of the circular side.
Thus; Girth = 2πr
- If length of cylinder is h, then we have;
2πr + h = 144
h = 144 - 2πr
- Now, to find the dimensions at which the max volume can be sent;
Volume of cylinder; V = πr²h
Let us put 144 - 2πr for h to get;
V = πr²(144 - 2πr)
V = 144πr² - 2π²r³
Differentiating with respect to r gives;
dV/dr = 288πr - 6π²r²
- Radius for max volume will be when dV/dr = 0
Thus; 288πr - 6π²r² = 0
Add 6π²r² to both sides to get;
288πr = 6π²r²
Rearranging gives;
288/6 = (π²r²)/πr
48 = πr
r = 48/π inches
- Put 48/π for r in h = 144 - 2πr to get;
h = 144 - 2π(48/π)
h = 144 - 96
h = 48 inches
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