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प्रश्न
The radius of a solid cylinder decreases by 10% and its height increases by 20%. Find the change in percentage of its volume and curved surface area
उत्तर
Initial values:
radius = r
height = h
Volume (V1) = πr2h
Curved surface area(C1) = 2πrh
New values after change :
radius
= r - 10% of r
= r - 0.1r
= 0.9r
height
= h + 20% of h
= h + 0.2h
= 1.2h
Volume(V2)
= π(0.9r)2 x 1.2h
= 0.972πr2h
Curved surface area(C2)
= 2π(0.9r)(1.2h)
= 2πrh(1.08)
Change in perenctage of Volume
= `(("V"_1 - "V"_2))/("V"_1) xx 100`
= `((pi"r"^2"h" - 0.972pi"r"^2"h"))/(pi"r"^2"h") xx 100`
= `(pi"r"^2"h"(1 - 0.972))/(pi"r"^2"h") xx 100`
= 0.028 x 100
= 2.8%
The positive value indicates that V1 is greater than V2(V1 - V2), which indicates that there is a decrease in volume by 2.8%. ...(Ans 1)
Change in the percentage of the
Curved surface area
= `(("C"_1 - "C"_2))/("C"_1) xx 100`
= `((2pi"rh" - 2pi"rh" xx 1.08))/(2pi"rh") xx 100`
= `(2pi"rh"(1 - 1.08))/(2pi"rh") xx 100`
= -0.08 x 100
= 8% ...(negative sign indicates that C1 is smaller than C2)
⇒ There is an 8% increase in the curved surface area. ...(Ans 2)
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