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Two similar cones have volumes 12π cu. units and 96π cu. units. If the curved surface area of the smaller cone is 15π sq. units, then what is the curved surface area of the larger cone?

Answer
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Hint: Here, we need to find the curved surface area of the larger cone. We will use the fact that the ratio of the heights and radii of two similar cones is equal. We will write the radius and height of one cone in terms of the radius and height of the other cone. Then, we will use the given information, formula for volume, formula for slant heights, and finally, the formula for the curved surface area of a cone to find the required area.

Formula used:
 We will use the following formulas:
1.The volume of a cone is given by the formula 13πr2h, where r is the radius of the base of the cone, and h is the height of the cone.
2.The slant height of a cone is given by the formula l=r2+h2, where r is the radius of the base of the cone, and h is the height of the cone.
3.The curved surface area of a cone is given by the formula πrl, where r is the radius of the base of the cone, and l is the slant height of the cone.

Complete step-by-step answer:
Let the height of the smaller and larger cone be h1 and h2 units respectively.
Let the radius of the smaller and larger cone be r1 and r2 units respectively.
So, now we will draw the diagrams of the cone.
seo images


It is given that the two cones are similar. This means that the ratios of the dimensions of the cones are equal.
Thus, the ratio of the heights of the cones is equal to the ratio of the radii of the cone.
Therefore, we get
r1r2=h1h2
Let r1r2=h1h2=n.
Therefore, we get the equations
r1=nr2 and h1=nh2
Now, we will calculate the volumes of the two cones.
Substituting r=r1 and h=h1 in the formula for volume of a cone 13πr2h, we get
Volume of the smaller cone =13π(r1)2h1
Substituting r1=nr2 and h1=nh2 in the equation, we get
Volume of the smaller cone =13π(nr2)2nh2
Simplifying the expression, we get
Volume of the smaller cone =13πn2(r2)2nh2=13n3π(r2)2h2
Substituting r=r2 and h=h2 in the formula for volume of a cone 13πr2h, we get
Volume of the larger cone =13π(r2)2h2
It is given that the volume of the smaller and larger cone is 12π cu. units and 96π cu. units respectively.
Thus, we get the equations
13n3π(r2)2h2=12π
and
13π(r2)2h2=96π
We will divide the two equations for volumes to form an equation in terms of n.
Dividing the equation 13n3π(r2)2h2=12π by the equation 13π(r2)2h2=96π, we get
13n3π(r2)2h213π(r2)2h2=12π96π
Simplifying the expression, we get
n3=18
Taking the cube root of both sides of the equation, we get
n=12
Substituting n=12 in the equation r1=nr2, we get
r1=12r2
Substituting n=12 in the equation h1=nh2, we get
h1=12h2
Now, we will calculate the slant heights of the two cones.
Let the slant height of the smaller and larger cone be l1 and l2 units respectively.
The slant height of a cone is given by the formula l=r2+h2, where r is the radius of the base of the cone, and h is the height of the cone.
Substituting l=l1, r=r1 and h=h1 in the formula for slant height, we get
l1=r12+h12
Substituting l=l2, r=r2 and h=h2 in the formula for slant height, we get
l2=r22+h22
Now, we will calculate the curved surface areas of the two cones.
The curved surface area of a cone is given by the formula πrl, where r is the radius of the base of the cone, and l is the slant height of the cone.
Substituting l=l1 and r=r1 in the formula for curved surface area of a cone, we get
Curved surface area of smaller cone =πr1l1
Substituting l1=r12+h12 in the equation, we get
Curved surface area of smaller cone =πr1r12+h12
It is given that the curved surface area of the smaller cone is 15π sq. units.
Thus, we get the equation
πr1r12+h12=15π
Substituting r1=12r2 and r1=12r2 in the equation, we get
π(12r2)(12r2)2+(12h2)2=15π
Simplifying the expression, we get
π(12r2)14r22+14h22=15ππ(12r2)14(r22+h22)=15ππ(12r2)12r22+h22=15π
Thus, we get
14πr2r22+h22=15π
Multiplying both sides of the equation by 4, we get
πr2r22+h22=60π
Substituting l=l2 and r=r2 in the formula for curved surface area of a cone, we get
Curved surface area of larger cone =πr2l2
Substituting l2=r22+h22 in the equation, we get
Curved surface area of larger cone =πr2r22+h22
Substituting πr2r22+h22=60π in the equation, we get
Curved surface area of larger cone =60π
Therefore, we get the curved surface area of the larger cone as 60π sq. units.

Note: We have not substituted π=227 because the volumes of the two cones, and the curved surface area of the smaller cone is given in the question in terms of π. Therefore, substituting the value π=227 and multiplying the expression is not needed.
We calculated the curved surface area of the cone in the question. The curved surface area of a cone is the area of the rounded surface of a cone. It is equal to the difference in the total surface area of the cone, and the area of the circular base of the cone.