Calculus w/ Analytic Geometry (chapter 1.5 question 33)

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Joe Mercurio

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A cylinder is inscribed in a sphere with fixed radius "a". If "H" is the height and "r" is the radius of the cylinder, express its volume and total surface area as a function of "H".

Surface Area = (2*pi*r2)+(2*pi*r*H)

r2 = a2-h2.
h=1/2*H
r2 = a2-(1/4H2)
Subbing this into the equation above:

Surface Area = 2*pi*[a2-1/4H2] + 2*pi*H*[SQRT(4a2-H2)]
Surface Area = 1/2*pi*[4a2 - H2] + 2*pi*H*[SQRT(4a2-H2)]

HOWEVER, the textbook answer is as follows:
Surface Area = 1/2*pi*[4a2-H2] + pi*H*[SQRT(4a2-H2)]

The difference being that the second half of their equation is not multiplied by 2 like my results are, and I can't figure out why they did that. It seems like they subbed in h=1/2H, but I don't see a reason to do that. Any help?

EDITED for typo in problem statement
 
Last edited:
Joe Mercurio said:
A cylinder is inscribed in a sphere with fixed radius "a". If "h" is the height and "r" is the radius of the cylinder, express its volume and total surface area as a function of "H".

Surface Area = (2*pi*r2)+(2*pi*r*H)

r2 = a2-h2.
h=1/2*H
r2 = a2-(1/4H2)
Subbing this into the equation above:

Surface Area = 2*pi*[a2-1/4H2] + 2*pi*H * [SQRT(4a2-H2)] ← that should be \(\displaystyle \sqrt{a^2 - \frac{H^2}{4}}\) which substitutes r
Surface Area = 1/2*pi*[4a2 - H2] + 2*pi*H*[SQRT(4a2-H2)]

HOWEVER, the textbook answer is as follows:
Surface Area = 1/2*pi*[4a2-H2] + pi*H*[SQRT(4a2-H2)]

The difference being that the second half of their equation is not multiplied by 2 like my results are, and I can't figure out why they did that. It seems like they subbed in h=1/2H, but I don't see a reason to do that. Any help?
Click to expand...

In your problem statement, you have If "h" is the height - then you want express its volume and total surface area as a function of "H".

Need to be more careful......
 
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