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Question

cylinder and cone have equal radius of their bases abd equal height,if their curved surface areas are in the ratio 8:5.show the radius and the height of each has the ratio 3:4

Answer 1

radius of each base = R
height of each = H
curved area of cylinder = 2πRH
curved area of cone = πRS
where S = distance from point of cone to circumference of the base. (slant height)
and
S^2 = R^2 + H^2
so if
2πRH / πRS = 8/5
then 2H/S = 8/5
10H = 8S
100H^2 = 64S^2 = 64(R^2 + H^2)
100H^2 = 64R^2 + 64H^2
36H^2 = 64R^2
36/64 = R^2/H^2
and
6/8 = 3/4 = R/H

Answer 2

The lateral surface area of a cylinder having height h and base radius r is base perimeter*height

The lateral surface area of a right cone having height h and base radius r is (1/2)*base perimeter*slant height

Thus, if the cylinder and the cone have equal base radii and equal heights, then the ratio of their lateral surface areas is:

(base perimeter*height)/(0.5**base perimeter*slant height)
=2*h/(slant height)

Assuming that the cone is a right circular cone, the slant height is the hypotenuse of a right triangle having legs of length h and r. So:

slant height^2 = h^2 + r^2
slant height = sqrt(h^2+r^2)

Therefore, the ratio of the surface areas is:

=2*h/sqrt(h^2+r^2)

This is equal to 8/5.

Take the unknown ratio of the radius to the height to be k. If:

r/h = k

then

r = kh.

Substitute this into the formula above:

=2*h/sqrt[h^2+(kh)^2]
=2*h/sqrt(h^2 + k^2h^2)
=2*h/sqrt[h^2(1 + k^2)]
=2*h/[h*sqrt(1+k^2)]
=2/sqrt(1+k^2)

This ratio is still equal to 8/5:

2/sqrt(1+k^2) = 8/5

In a proportion, the product of the means is equal to the product of the extremes.

2*5 = 8*sqrt(1+k^2)

10 = 8*sqrt(1+k^2)

100 = 64(1 + k^2)

100/64 = 1 + k^2

100/64 - 1 = k^2

100/64 - 1 = 100/64 - 64/64 = 36/64 = 9/16

So k^2 = 9/16. Take the square root of both sides, and k must be 3/4. We are only interested in the principal square root, since the ratio of positive lengths cannot be negative:

Cheers!

Answer 3

Guess you want the proof.
R=base radius
h= height
l= slant length of cone

curved surface area of cylinder= 2 pi R h
curved surface area of cone= pi R l

since h and R are equal for both, and the ratios are 8:5, we can write an equation-->
2 pi R h = 8/5 pi R l
cancelling pi & R on both sides, we have
2 h = 8/5 l
=> h= 4/5l
=> h^2= 16/25 l^2
=> 25/16h^2= l^2

now, l^2 = h^2 + R^2 (pythagoras theorem)
or, l = sq root (h^2 + R^2)

so, 25/16h^2= h^2 + R^2
or, (25-16)/16h^2 = R^2
or, 9/16h^2 = R^2
h^2=16/9R^2
h= 4/3 R
or, 3h= 4R
or, R:h :: 4:3

Answer 4

Surface areas are 2*pi*r*h and pi*r*l, where
l = sqrt(r^2 + h^2) . Taking ratios of the above area expressions and using the given information, we get
2*h:l = 8:5, and so
10*h = 8*l. Square this and use the above equation for l, getting
100h^2 = 64*(r^2 + h^2)

and so 36h^2 = 64r^2
Thus 6h = 8r and so
r:h = 6:8 = 3:4

Answer 5

formula for curved surface area of cylinder - 2 x 3.142 x r x h (where h is the vertical height and r is the radius and 3.142 is pie)

formula for curved surface area of cone - 3.142 x r x h (where h is the slant height)

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