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If we know just the surface area $A$, and the slant height $h_s$ is there a way to find the radius $r$ of the base of a cone?

The surface area formula for cones is:

$$A = ( \pi \cdot r \cdot h_s ) + ( \pi \cdot r^2 ) $$

So if we have this problem:

$$( 3.14 \cdot r \cdot 10 ) + ( 3.14 \cdot r^2 ) = 175.84 $$

Does anyone know how to find $r$?

$r$ should equal 4.

I'm just not sure of the method to get there.

Sorry this has taken me a whole day to try to figure out, even though I have all the other formulas for surface area, and volume figured out, with missing variables. This is the only one that has made me confused lol.

I've tried more then 10 different ways, including trying to simplify it, and replace the rest of the operational signs with their opposites, but that does not give the correct answer.

Thanks for any help you can give!

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  • $\begingroup$ Thank you Cold Number for your helpful edit! $\endgroup$ – Digital-Christie Sep 4 '15 at 22:15
  • $\begingroup$ You're welcome! $\endgroup$ – coldnumber Sep 4 '15 at 22:20
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It's a quadratic equation in terms of $r$, so you can find $r$ using the quadratic formula.

First, rewrite it as $$ 0=\pi r^2+10\pi r - 175.84$$

Then use the quadratic formula:

$$r=\frac{-10\pi \pm \sqrt{(10\pi)^2-4\pi(-175.84)}}{2\pi}$$

Then you can just simplify.


As a side note, if you are asked for the answer to two decimal places, it's better to leave the rounding until the very end; the earlier you start rounding, the more different your approximation will be from the exact answer. (So don't turn $\pi$ into $3.14$ before simplifying the quadratic formula.)

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    $\begingroup$ This is it! Thank you! =) Yes simply using 3.14 in this equation gave me the wrong answer (8.29) but using the long Pi number (3.14159265359) gave me the right answer. (4.000450719) I'd rate this answer up if I had more reputation. $\endgroup$ – Digital-Christie Sep 4 '15 at 23:06
  • $\begingroup$ I'm glad this helped :) $\endgroup$ – coldnumber Sep 4 '15 at 23:31
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Well, we have:

$$3.14 \cdot r^2 + 31.4 \cdot r = 175.84$$

Rearranging delivers:

$$3.14 \cdot r^2 + 31.4 \cdot r - 175.84 = 0$$

This is a quadratic equation in $r$, thus

$$r = \frac{-31.4 \pm \sqrt{31.4^2 - 4 \cdot 3.14 \cdot (-175.84)}}{2 \cdot 3.14}$$

which leads to

$$r_1 = 4 \qquad \text{and} \qquad r_2 = -14$$

but $r > 0$, so $r_1$ is the solution.

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  • $\begingroup$ Thank you so much for giving an answer! Though is this equation missing the slant height, which is 10 in my example? $\endgroup$ – Digital-Christie Sep 4 '15 at 23:09
  • $\begingroup$ Note that I've already done the multiplication $3.14 \cdot 10 = 31.4$ $\endgroup$ – root Sep 4 '15 at 23:26
  • $\begingroup$ Oh of coarse! Your right! ^_^ Thank you! $\endgroup$ – Digital-Christie Sep 4 '15 at 23:29
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Assuming we're talking about a right circular cone, the inside contains a right triangle consisting of the radius and height as the sides, and the slant height as the hypotenuse (which I'll call "x" in the calculation below):

$$r^2 + h^2 = x^2 \ \ or \ \sqrt{r^2 + h^2} = x$$

given that the equation for surface area of a cone is: $$A = πr (r+\sqrt{h^2+r^2})$$

we can simply replace the$ \sqrt{r^2 + h^2}$ part of the surface area equation with the value for x. The surface area equation finally simplifies to:

$$175.84=\pi r^2 + 10\pi r $$

move the left hand side to the right, then use the quadratic equation to solve. You should come up with two answers.

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    $\begingroup$ Awesome! Thank you for your help! Yes it comes down to the quadratic formula. I got the correct answer with that! Yes I thought of using the pythagorean theorem, but I couldn't figure out how to get the perpendicular height. $\endgroup$ – Digital-Christie Sep 4 '15 at 23:14

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