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Calculus/Calculus optimization
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Question
Find the dimensions of a right circular cone with surface area 1 and maximal volume. Answers in simplest radical form. Please help me with this question for I'm so so so stuck. Thanks a lot :)
The surface area of a cone involves the top, the bottom, and the side.
The top and bottom are circles. This means the area of each is πr².
Since there are two, the total area is 2πr².
The side is a rolled up rectangle. The height is the height of the cone
and the base is the circumference of a circle of radius r.
The circumference of the circle is 2πr, and the height is h, so that is 2πrh.
This gives the total surface area as 2πr² + 2πrh, which is said to be 1.
The volume of a cone is known to be V = πr²h.
The total surface area was given as 1 = 2πr² + 2πrh.
Using the surface area to find h gives h = (1 - 2πr²)/(2πr).
This can be put in V to give V(r) = πr²(1 - 2πr²)/(2πr).
Multiplying the top out, find V'(r) with the quotient rule, and set it to 0.
Solve for r. Use h = (1 - 2πr²)/(2πr) to find h.
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