find the smallest surface area of a cylinder holding a volume of 300 cubic cm.
+2498 \(V_{cylinder}= \pi r ^2 h \\ 300=\pi r^2h\\ h=\frac{300}{\pi r^2} \)
so let s put our h=300/(pi r^2) in \(S_{cylinder}=2 \pi rh+2 \pi r^2 \)
\(S_{cylinder}=2 \pi r( \frac{300}{\pi r^2 })+2 \pi r^2\\ S_{cylinder}=\frac{600}{r}+2 \pi r^2 \\ y=\frac{600}{x}+2\pi x^2 \text{ (so the minimum value of x is 3.628) }\)
https://www.desmos.com/calculator/mae20kmdjb
so r is 3.628 put it in term s of \(h=\frac{300}{\pi r^2}\)
\(h=\frac{300}{3.628^2 \pi} \\ h = 7.25499\\ S_{cylinder}=2 \times \pi \times 3.628 \times 7.25499+2\times\pi\times 3.628^2=248.0820699 \dots\)
+2498 \(V_{cylinder}= \pi r ^2 h \\ 300=\pi r^2h\\ h=\frac{300}{\pi r^2} \)
so let s put our h=300/(pi r^2) in \(S_{cylinder}=2 \pi rh+2 \pi r^2 \)
\(S_{cylinder}=2 \pi r( \frac{300}{\pi r^2 })+2 \pi r^2\\ S_{cylinder}=\frac{600}{r}+2 \pi r^2 \\ y=\frac{600}{x}+2\pi x^2 \text{ (so the minimum value of x is 3.628) }\)
https://www.desmos.com/calculator/mae20kmdjb
so r is 3.628 put it in term s of \(h=\frac{300}{\pi r^2}\)
\(h=\frac{300}{3.628^2 \pi} \\ h = 7.25499\\ S_{cylinder}=2 \times \pi \times 3.628 \times 7.25499+2\times\pi\times 3.628^2=248.0820699 \dots\)
+118667 Good work Solveit,
I have not checked it thoroughly but it looks good.
I would suggest that you use calculus to find the minimum rather than using a graph generated by Desmos.
Do you know how to do that?
+129847 Here's the Calculus approach.........first, solve the volume in terms of h
300 = pi*r^2 *h
h = [300] /[pi*r^2] .......now....substitute this into the surface area "formula"
Sarea = 2pi*r^2 + 2pi*r*h
Sarea = 2pi*r^2 + 2pi*r* [300]/[pi*r^2]
Sarea = 2pi*r^2 + 600r^(-1) ....take the derivative with respect to r......
S' area = 4pi*r - 600r^(-2) .....set this to 0
4pi*r - 600r^(-2) = 0
4pi*r = 600r^(-2)
pi*r = 150r^(-2)
r^3 = 150/pi
r = (150/pi)^(1/3) = about 3.628 .....as Solveit found .......
And h = 300/ [pi * 3.628^2] = about 7.255
And the minimum surface area would be just as Solveit found......
