Optimisation: Perimeter of window

[Monkeyseat]

Hello, the main trouble I have with Optimisation is getting the formula.

Here is the question (I am on 3aii at the moment):

Here is my working:

I can't seem to get the same as the book, where am I going wrong?

Thanks. Sorry for posting it as an image but I find it easier to write it out than type it.

 

[wjm11]

Hello, Monkeyseat,

I suspect there are two problems here. First, I’m guessing that the 500 cm^2 only applies to the rectangular area, not the hemisphere. Second, I think the book has an error; the perimeter formula should not have a 2 under the pi term. If my guesses are correct, then

h = 500/2r

p = 2r + pi*r + 2h = (2 + pi)r + 500/r

 

[Monkeyseat]

Hello, Monkeyseat,

I suspect there are two problems here. First, I’m guessing that the 500 cm^2 only applies to the rectangular area, not the hemisphere. Second, I think the book has an error; the perimeter formula should not have a 2 under the pi term. If my guesses are correct, then

h = 500/2r

p = 2r + pi*r + 2h = (2 + pi)r + 500/r

I'm not sure about the area. I got section (a) correct about the formula connecting h and r meaning 2hr + (pi r^2)/2 = 500 so I presumed that 500cm^2 is the area if the whole thing.

I think pi/2 is correct because that is needed to get the correct answer to part b(i) which I have just done using the answer I am meant to get for a(ii). I used pi/2 in part b(i) and got the same answer as the book so I believe pi/2 is needed.

Thanks for replying but any more ideas?

 

[soroban]

Hello, Monkeyseat!

You have the correct function . . . you just didn't recognize it.


\(\displaystyle \text{You had: }\ \;=\;\left(2 - \frac{\pi}{2}\right)r + \frac{500}{r} + \pi r\)


\(\displaystyle \text{This is: }\;P \;=\;2r - \frac{\pi}{2}r + \frac{500}{r} + \pi r \;=\;2r + \frac{\pi}{2}r + \frac{500}{r} \;=\;\left(2 + \frac{\pi}{2}\right)r + \frac{500}{r}\)
. . . . . . . . . . . . . . . . . . . . . \(\displaystyle \nwarrow\qquad\nearrow\)
. . . . . . . . . . . . . . . . . . . . . . \(\displaystyle \text{combine}\)

 

[Monkeyseat]

Hello, Monkeyseat!

You have the correct function . . . you just didn't recognize it.


\(\displaystyle \text{You had: }\ \;=\;\left(2 - \frac{\pi}{2}\right)r + \frac{500}{r} + \pi r\)


\(\displaystyle \text{This is: }\;P \;=\;2r - \frac{\pi}{2}r + \frac{500}{r} + \pi r \;=\;2r + \frac{\pi}{2}r + \frac{500}{r} \;=\;\left(2 + \frac{\pi}{2}\right)r + \frac{500}{r}\)
. . . . . . . . . . . . . . . . . . . . . \(\displaystyle \nwarrow\qquad\nearrow\)
. . . . . . . . . . . . . . . . . . . . . . \(\displaystyle \text{combine}\)

Ohhhhh thanks! I thought about something like that but I didn't know you could do that.

Is it basically like doing the following: (-0.5*pi*r) + (1*pi*r) = 0.5 pi r

Or even more basic without the constants (which would be put in, 0.5 is not the answer just the number infront of the pi and r): -0.5 + 1 = 0.5?

Just wondering if that is right, is it?

Thanks.