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top11 2021-5-14 17:45

[quote]原帖由 [i]小朗朗[/i] 於 2021-5-12 04:28 PM 發表 [url=https://www.discuss.com.hk/redirect.php?goto=findpost&pid=535305528&ptid=29897076][img]https://www.discuss.com.hk/images/common/back.gif[/img][/url]
12337657 [/quote]
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(a)
By similar triangles, the upper cone has a height to radius ratio of 20:10 = 2:1.
Therefore, if the cylinder has a radius of r cm, then the height is (20 - 2r) cm.

Its volume is V = πr²(20 - 2r) cm³.
When r = 7, V = π (49) (20 - 14) = 294π
When r = 9, V = π (81) (20 - 18) = 162π

Note that dV/dr
= d/dr πr²(20 - 2r)
= d/dr π(20r² - 2r³)
= π(40r - 6r²)
= 2πr(20 - 3r) > 0 if 0 < r < 20/3

dV/dr < 0 for r > 20/3 ≈ 6.6667

Therefore, if the base radius of the cylinder increases from 7 cm to 9 cm, the volume of the cylinder decreases from 294π cm³ to 162π cm³.

(b)
From the steps in (a) above, maximum V occurs at r = 20/3.
The maximum volume is π (20/3)² (20 - 40/3) = 8000π/27 cm³.