Answer:
radius r = 47/Ï€ in
length l = 23.5 in
Step-by-step explanation:
Let us recall that:
Volume of a cylinder = [tex]\pi r^2 h[/tex] --- (1)
The girth g = [tex]2 \pi r + h[/tex] Â Â Â ----- (2)
Given that:
height (h) + girth (g) = 141
Then g = 141 - h  ----- (3)
Equating equation (2) and (3), we have:
141 - h = 2Ï€r + h
2h = 141 - 2Ï€r
h = 70.5 - πr    ------ (4)
From, here now, we can now replace the value of h into equation (1)
i.e.
V = πr²(70.5 - πr)
V = 70.5πr² -  π²r³
Taking the differential of the above equation with respect to r, we have:
[tex]\dfrac{dv}{dr} = 141 \pi r- 3\pi ^2 r^2[/tex]
By further differentiation:
[tex]\dfrac{d^2v}{dr^2} = 141 \pi - 6\pi ^2 r[/tex]
Let set [tex]\dfrac{dv}{dr}= 0[/tex], Then:
141πr - 3π²r² = 0
141πr = 3π²r²
Divide both sides by πr
141 = 3Ï€r
r = 141/3Ï€
r = 47/Ï€ in
Replacing the value of r = 47/Ï€ into equation (4), we have:
h = 70.5 - πr
h = 70.5 - π(47/π)
h = 70.5 - 47
h = 23.5 in
From equation (3);
h + g = 141
23.5 + g = 141
g = 141 - 23.5
g = 117.5 in
Volume = πr²h
V = π × (47/π )² × 23.5
V = 16523.94 in³
