Differentiation - Stationary Points (A cone of radius r and height h has volume 81pi)

[sojeee]

A cone of radius r and height h has volume 81pi.
a) Show that the curved surface area of the cone is given by S= pi x sqrt r⁴ + 243²/r²

My working:

Volume = 1/3 x pi x r²x h
81pi = 1/3 x pi x r²x h
243/r²= h

Surface Area = pi x r x l

l² = h² + r²
l^2 = (243/r^2)^2 + r^2
l^2 = 59049/r^4 +r^2
l^2 = 59094/r^4 +r^6/r^4
l^2 = 59094 + r^6/ r^4
l = sqrt all the above
l = sqrt59094 + r^6/ r^2

S = pi x r x l
= pi x r x sqrt59094 + r^6/ r^2
= pi x sqrt 59094 + r^6/ r

No idea how to get to the answer they want after this point.

 

[Dr.Peterson]

A cone of radius r and height h has volume 81pi.
a) Show that the curved surface area of the cone is given by S= pi x sqrt(r⁴ + 243²)/r²

My working:

Volume = 1/3 x pi x r²x h
81pi = 1/3 x pi x r²x h
243/r²= h

Surface Area = pi x r x l

l² = h² + r²
l^2 = (243/r^2)^2 + r^2
l^2 = 59049/r^4 + r^2
l^2 = 59094/r^4 + r^6/r^4
l^2 = (59094 + r^6)/ r^4
l = sqrt all the above
l = sqrt(59094 + r^6)/ r^2

S = pi x r x l
= pi x r x sqrt(59094 + r^6)/ r^2
= pi x sqrt(59094 + r^6)/ r

No idea how to get to the answer they want after this point.

I added parentheses above so that what you wrote means what you must have intended. You also often typed 59094 where you meant 59049.

Note that you got 59049 by squaring 243, so let's write your result so far more like they did:

S= pi x sqrt(243^2 + r^6)/r

Now, their answer differs from yours only in the exponents. Clearly they can't be equal, so we have to look for an error in your work (or maybe in theirs).

I don't see an error in your work. And looking at the units involved, I see that 243^2 has units of volume^2, or m^6, so it makes sense to add it to r^6; the square root results in m^3, and dividing by r leaves you with an area. Their answer doesn't work out.

Check whether you copied it correctly! (Or did I misinterpret it?)

 

[sojeee]

I added parentheses above so that what you wrote means what you must have intended. You also often typed 59094 where you meant 59049.

Note that you got 59049 by squaring 243, so let's write your result so far more like they did:

S= pi x sqrt(243^2 + r^6)/r

Now, their answer differs from yours only in the exponents. Clearly they can't be equal, so we have to look for an error in your work (or maybe in theirs).

I don't see an error in your work. And looking at the units involved, I see that 243^2 has units of volume^2, or m^6, so it makes sense to add it to r^6; the square root results in m^3, and dividing by r leaves you with an area. Their answer doesn't work out.

Check whether you copied it correctly! (Or did I misinterpret it?)

i think you have misinterpreted the question. As in the parentheses is in the wrong place in the question they gave. It should be like this S = pi x sqrt [ r^4 + (243^2/r^2) ]. Sorry for the confusion

 

[Dr.Peterson]

i think you have misinterpreted the question. As in the parentheses is in the wrong place in the question they gave. It should be like this S = pi x sqrt [ r^4 + (243^2/r^2) ]. Sorry for the confusion

Actually, I realized shortly after I wrote that that must be the issue.

Do you see yet how to obtain that formula?

Just take S= pi sqrt(243^2 + r^6)/r and change it to pi sqrt(243^2 + r^6)/sqrt(r^2), then combine the radicals into one.