89.4k views
7 votes
the volume of two similar solids are 1080cm and 1715cm .if the curved surface area of the smaller cone is 840cm .fond the curved surface area of the larger cone​

User StfBln
by
7.3k points

1 Answer

4 votes

Answer:


A_(big) = 1143.33cm^2

Step-by-step explanation:

The given parameters are:


V_(small) = 1080


V_(big) = 1715


C_(small) = 840

Required

Determine the curved surface area of the big cone

The volume of a cone is:


V = (1)/(3)\pi r^2h

For the big cone:


V_(big) = (1)/(3)\pi R^2H

Where

R = radius of the big cone and H = height of the big cone

For the small cone:


V_(small) = (1)/(3)\pi r^2h

Where

r = radius of the small cone and H = height of the small cone

Because both cones are similar, then:


(H)/(h) = (R)/(r)

and


(V_(big))/(V_(small)) = ((1)/(3)\pi R^2H)/((1)/(3)\pi r^2h)


(V_(big))/(V_(small)) = (R^2H)/(r^2h)

Substitute values for Vbig and Vsmall


(1715)/(1080) = (R^2H)/(r^2h)

Recall that:
(H)/(h) = (R)/(r)

So, we have:


(1715)/(1080) = (R^2*R)/(r^2*r)


(1715)/(1080) = (R^3)/(r^3)

Take cube roots of both sides


\sqrt[3]{(1715)/(1080)} = (R)/(r)

Factorize


\sqrt[3]{(343*5)/(216*5)} = (R)/(r)


\sqrt[3]{(343)/(216)} = (R)/(r)


(7)/(6) = (R)/(r)

The curved surface area is calculated as:


Area = \pi rl

Where


l = slant\ height

For the big cone:


A_(big) = \pi RL

For the small cone


A_(small) = \pi rl

Because both cones are similar, then:


(L)/(l) = (R)/(r)

and


(A_(big))/(A_(small)) = (\pi RL)/(\pi rl)


(A_(big))/(A_(small)) = (RL)/(rl)

This gives:


(A_(big))/(A_(small)) = (R)/(r) * (L)/(l)

Recall that:


(L)/(l) = (R)/(r)

So, we have:


(A_(big))/(A_(small)) = (R)/(r) * (R)/(r)


(A_(big))/(A_(small)) = ((R)/(r))^2

Make
A_(big) the subject


A_(big) = ((R)/(r))^2 * A_(small)

Substitute values for
(R)/(r) and
A_(small)


A_(big) = ((7)/(6))^2 * 840


A_(big) = (49)/(36) * 840


A_(big) = (49* 840)/(36)


A_(big) = 1143.33cm^2

Hence, the curved surface area of the big cone is 1143.33cm^2

User Italktothewind
by
7.1k points