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Q.

The cost of painting the total surface area of cones at 25 paise per c m 2   is Rs. 176 and slant height is 25 cm then the Volume of cone is ?


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a

550 c m 3  

b

528 c m 3  

c

666 c m 3  

d

429 c m 3   

answer is B.

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Detailed Solution

We have drawn the figure of the given right circular cone above with apex A and the centre of circular base B and C a point on the circular base. The line segment joining the apex to the centre is the height of the cone denoted as h and the line segment joining the apex to any point the circular base is called slant height and denoted as  l  . The surface made by the slant height is called the curved surface of the cone. The total surface area with radius at the base r is given by
TSA=π r 2 +πrl……(1)  
The volume of the cone is given by
V=π r 2 h=π r 2 l 2 − r 2  
We give in the question that the cost of painting the total surface area of cone at 25 paisa per cm 2   is Rs 176 and the slant height is l=25 cm  .
So the total surface area is
TSA=  total cost  cost  per cm 2 = 176 0.25 =704  cm 2 ……(2)  
We put l=25 cm  and equate the right hand sides of equation (1) and (2) to have;
⇒π r 2 +πr×25=704 ⇒π r 2 +25r =704  
⇒ r 2 +25r−224=0  
The above equation is a quadratic equation in r  which we solve by splitting the term 25r  . We have;
⇒ r 2 +32r−7r−224=0 ⇒r(r+32)−7(r+32)=0 ⇒(r+32)(r−7)=0 ⇒r+32=0 or r−7=0  
So we have two solutions to the quadratic equation r=−32,7  . We reject r=−32  since length cannot be negative and hence conclude r=7 cm  . We use the formula for volume of a cone and find the required volume with
V= 1 3 π r 2 l 2 − r 2 V= 1 3 22 7 ×7×7× 25 2 − 7 2 V= 1 3 ×22×7×24=528  cm 3  
 
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