SOLUTION: The radius of the base of each of two right circular cones is 8 inches. The altitude of the first cone is 9 greater than that of the second cone and its slant height is 7 inches gr

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Question 900733: The radius of the base of each of two right circular cones is 8 inches. The altitude of the first cone is 9 greater than that of the second cone and its slant height is 7 inches greater than that of the second. Find the altitude of the taller cone.
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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The radius of the base of each of two right circular cones is 8 inches.
The altitude of the first cone is 9 greater than that of the second cone
and its slant height is 7 inches greater than that of the second.
Find the altitude of the taller cone.
:
This can be done by considering two right triangles, using pythag formulas
let x = the height of taller cone
then
(x-9) = the height of the smaller cone
The slant dimensions are the hypotenuses
sqrt%28x%5E2%2B8%5E2%29 - 7 = sqrt%28%28x-9%29%5E2+%2B+8%5E2%29
sqrt%28x%5E2%2B64%29 - 7 = sqrt%28%28x%5E2-18x%2B81%29%2B64%29
sqrt%28x%5E2%2B64%29 - 7 = sqrt%28%28x%5E2-18x%2B145%29%29
square both sides
x%5E2+%2B+64+-+14sqrt%28x%5E2%2B64%29+%2B+49+=+x%5E2+-+18x+%2B+145
subtract x^2 from both sides
64+-+14sqrt%28x%5E2%2B64%29+%2B+49+=+-+18x+%2B+145
-14sqrt%28x%5E2%2B64%29+%2B+113+=+-+18x+%2B+145
-14sqrt%28x%5E2%2B64%29+=+-+18x+%2B+145+-+113
-14sqrt%28x%5E2%2B64%29+=+-18x+%2B+32
simplify, change the signs, divide by -2
7sqrt%28x%5E2%2B64%29+=+9x+-+16
square both sides
49(x^2 + 64) = 81x^2 - 288x + 256
49x^2 + 3136 = 81x^2 - 288x + 256
combine like terms on the right
0 = 81x^2 - 49x^2 - 288x + 256 - 3136
0 = 32x^2 - 288x - 2880
simplify divide by 32
x^2 - 9x - 90 = 0
Factors to
(x-15)(x+6) =
the positive solution here
x = 15 in is the altitude of the taller cone
:
:
See if that checks out the height of the smaller will be 6
sqrt%2815%5E2%2B8%5E2%29 = 17
sqrt%286%5E2%2B8%5E2%29 = 10
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slant differences 7 in