Question

the total surface area of a right cone whose slant height is 25cm is704cm squared. calculate its radius

Answer 1

To find the radius of a right cone when given the slant height and the total surface area, we can use the formula for the total surface area of a cone, which is given by:

\[
SA = \pi r l + \pi r^2
\]

where \( SA \) is the total surface area, \( r \) is the radius, and \( l \) is the slant height.

Given:
- \( SA = 704 \, \text{cm}^2 \)
- \( l = 25 \, \text{cm} \)

Substituting these values into the formula, we get:

\[
704 = \pi r (25) + \pi r^2
\]

We can factor out \( \pi \):

\[
704 = \pi (25r + r^2)
\]

Next, divide both sides by \( \pi \):

\[
\frac{704}{\pi} = 25r + r^2
\]

Calculating \( \frac{704}{\pi} \):

\[
\frac{704}{\pi} \approx 224.0 \, \text{cm}^2
\]

Thus, we have the equation:

\[
224 = 25r + r^2
\]

Rearranging gives us:

\[
r^2 + 25r - 224 = 0
\]

Now, we can solve this quadratic equation using the quadratic formula:

\[
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = 25 \), and \( c = -224 \). Plug in these values:

\[
b^2 - 4ac = 25^2 - 4 \times 1 \times (-224) = 625 + 896 = 1521
\]

Calculating the square root:

\[
\sqrt{1521} = 39
\]

Now substitute back into the quadratic formula:

\[
r = \frac{-25 \pm 39}{2}
\]

Calculating the two possible solutions:

1.

\[
r = \frac{14}{2} = 7
\]

2.

\[
r = \frac{-64}{2} = -32 \quad \text{(not applicable, as radius cannot be negative)}
\]

Thus, the radius \( r \) is:

\[
\boxed{7 \, \text{cm}}
\]