F4 amath differentiation

(a)先設果個frustum做個一個完整既cone 先,就叫最尖個點做F
咁未個果個高設c
用similar triangle做比例
c/(c h)=a/b
...
c=ha/(b-a)


用成個cone FAB volume 減小cone FCD volume
1/3b^2(h c)&-1/3a^2c& ( &=3.14 )
1/3b^2[h ha/(b-a)]&-1/3a^2[ha/(b-a)]&
用通分母既方法
分子會出現b^3-a^3
分母係b-a
約簡就做到答案





(bi)用返part (a) 既volume
即係h&/3(a^2 ab b^2) (1)
h=2r (2)
r/[email protected]=r/tan@ (3)
a/[email protected]=rtan@ (4)

將2,3,4 sub into (1),抽佐common factor出黎之後會有一舊
(tan^2@ 1 1/tan^2@),then 變做(tan^2@ 2 1/tan^2@-1).....done







(bii) differentiate part (bi)個volume,會出佐舊好長既野dv/dt=
2&/3{[(tan@ 1/tan@)^2-1] 3r^2 dr/dt 2r^3(tan@ 1/tan@)(sec^2@-csc^2@)d@/dt}

之後小心d sub d數入去就計到,不過唔知點解我計到 -178






(biii) dr/dt constant ,即係話dr/dt=0
咁要volume minimum,依個sec^2@-csc^2@=0 因為tan@ 1/tan@ 不會=0

then sec^2@-csc^2@=0
tan@=1
@=&/4