# Math Help - What This Means: Ratio of the circle's radius to the height of the rectangle?

1. ## What This Means: Ratio of the circle's radius to the height of the rectangle?

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2. Originally Posted by AlphaRock
A window has the shape of a rectangle and the upper portion's shape is a semicircle connected with the rectangle's upper side.

The perimeter = 15 units = 2r+2l+(pi)r

What is the ratio of the circle's radius to the height of the rectangle so that the area of the window is a maximum?

I know how to get the maximum area. But I don't know what the "ratio of the circle's radius to the height of the rectangle" means.

$\frac{r}{l}$

3. Originally Posted by skeeter

$\frac{r}{l}$

Thanks for making it clear, skeeter.

Is that right?

4. $15 = 2L + (\pi+2)r$

$L = \frac{15 - (\pi+2)r}{2}$

$A = 2rL + \frac{\pi r^2}{2}$

$A = 15r - (\pi+2)r^2 + \frac{\pi r^2}{2}$

$A = 15r - \left(\frac{\pi}{2} + 2\right)r^2$

$\frac{dA}{dr} = 15 - (\pi + 4)r = 0$

$r = \frac{15}{\pi + 4}$

I get $\frac{r}{L} = 1$

5. Originally Posted by skeeter
$15 = 2L + (\pi+2)r$

$L = \frac{15 - (\pi+2)r}{2}$

$A = 2rL + \frac{\pi r^2}{2}$

$A = 15r - (\pi+2)r^2 + \frac{\pi r^2}{2}$

$A = 15r - \left(\frac{\pi}{2} + 2\right)r^2$

$\frac{dA}{dr} = 15 - (\pi + 4)r = 0$

$r = \frac{15}{\pi + 4}$

I get $\frac{r}{L} = 1$
Thanks! I see what I did wrong now.