A cone of height 24 cm has a curved surface area 550 cm^2. Find its volume. (take pi = 22/7).

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sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the volume of the cone using the following formula, such that:

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

The problem provides the height h and curved surface area, hence, you need to find radius using the formula of curved surface area, such that:

`CSA = pi*r*l`

l represents the slant height

Using Pythagorean theorem, you may evaluate l, such that:

`l^2 = h^2 + r^2 => l = sqrt(h^2 + r^2)`

`l = sqrt(24^2 + r^2)`

Substituting `sqrt(24^2 + r^2)` for l in formula of CSA, yields:

`550 = pi*r*sqrt(24^2 + r^2) `

`550^2 = 22^2/49*r^2(24^2 + r^2)`

`30625 = 576r^2 + r^4 => r^4 + 576r^2 - 30625 = 0`

Substituting t for `r^2` yields:

`t^2 + 576t - 30625 = 0`

Using quadratic formula yields:

`t_(1,2) = (-576 +- sqrt(454276))/2 => t_(1,2) = (-576 +- 674)/2`

`t_1 = 49, t_2 = -625`

`r^2 = 49 => r_1 = 7, r_2 = -7` invalid

`r^2 = -625` invalid

Substituting 7 for r in equation of volume yields:

`V = (1/3)*pi*r^2*h =>` ` V = (1/3)*(22/7)*49*24`

`V = 1232 cm^3`

Hence, evaluating the volume of the cone, under the given conditions, yields `V = 1232 cm^3.`

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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The volume of a cone with height h and base radius r is `V = (1/3)*pi*r^2*h` and the curved surface area is `CSA = pi*r*sqrt(r^2 + h^2)` .

For a cone with curved surface area 550 cm^2 and height 24 cm

`550 = (22/7)*r*sqrt(r^2 + 576)`

=> `175 = r*sqrt(r^2 + 576)`

take the square of both the sides

=> `30625 = r^2*(r^2 + 576)`

=> `r^4 + 576r^2 - 30625 = 0`

solving the equation gives 4 roots of which r = 7 is the only real and positive root.

The volume of the cone is `(1/3)*(22/7)*7^2*24 = 1232` cm^3

The volume of the cone is 1232 cm^3

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