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2.11.3 Differentiation (Paper 2), Question 4


Question 4:
(a) A worker is pumping air into a spherical shape balloon at the rate of  25 cm3 s-1.
[ Volume of a sphere, V= 4 3 π r 3 ]
Leaving answer in terms of π, find,
(i) rate of change of radius of the balloon when its radius is 10 cm.   [3 marks]
(ii) approximate change of volume when radius of the balloon decrease from 10 cm to  9.95 cm.    [2 marks]
(b) A curve y=h x 3 3 x  has turning point x = 1, find value of h.  [3 marks]

Solution:
(a)(i)
V= 4 3 π r 3 dV dr =4π r 2 dr dt = dr dV × dV dt = 1 4π r 2 ×25 = 1 4π ( 10 ) 2 ×25 = 1 16π

(a)(ii)
δV δr dV dr δV= dV dr ×δr =4π r 2 ×( 9.9510 ) =4π ( 10 ) 2 ×( 0.05 ) =20π

(b)
y=h x 3 3 x y=h x 3 3 x 1 dy dx =3h x 2 +3 x 2 dy dx =3h x 2 + 3 x 2 At turning points,  dy dx =0 0=3h x 2 + 3 x 2 0=3h ( 1 ) 2 + 3 1 given x=1 3h=3 h=1


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