# Thread: Optimization Problem with Triangles

1. ## Optimization Problem with Triangles

hey guys can you help me here.

A right triangle has area 50.
a) Express the hypotenuse as a function of a single variable, and
b) Find the shortest possible hypotenuse this triangle can have

I know that bh/2=50. They're asking me to write b^2+h^2=c^2 in terms of a single variable?

2. Originally Posted by RezMan
hey guys can you help me here.

A right triangle has area 50.
a) Express the hypotenuse as a function of a single variable, and
b) Find the shortest possible hypotenuse this triangle can have

I know that bh/2=50. They're asking me to write b^2+h^2=c^2 in terms of a single variable?
You're almost there:
bh/2 = 50 so
bh = 100
b = h/100 [MODERATOR EDIT]: See below for correction.
We want c = sqrt( b^2 + h^2 ), which becomes
c = sqrt( (h/100)^2 + h^2 )
You can rewrite that as needed.

b) Find c'.
Set it = 0.
etc

3. wow you make it seem so simple! I was wondering, doesn't sqrt(b^2+h^2) turn into just b+h. And isn't it 100/h not h/100. I can't thank you enough for your help!

4. Originally Posted by RezMan
wow you make it seem so simple! I was wondering, doesn't sqrt(b^2+h^2) turn into just b+h. Mr F says: And I suppose you also want to turn sqrt(1^2 + 1^2) into 1 + 1 ....

And isn't it 100/h not h/100. Mr F says: Correct.

I can't thank you enough for your help!
..

5. Originally Posted by mr fantastic
And I suppose you also want to turn sqrt(1^2 + 1^2) into 1 + 1
I guess not?
so what would I do with $\displaystyle \sqrt{\frac{10000}{h^2}+h^2}$? Should I take the derivative of that or could I reduce it more

6. Originally Posted by RezMan
I guess not?
so what would I do with $\displaystyle \sqrt{\frac{10000}{h^2}+h^2}$? Should I take the derivative of that or could I reduce it more
Yeah, maybe if someone would be so kind as to edit my mistake above...! Glad you could see the intended result...

There may be a nice trig substitution, but you can always just crank out the derivative...

7. I feel this is wrong but I don't know what else to do. ANyway. so Quotient rule?

$\displaystyle \sqrt{\frac{h^2(0)-10000(2h)}{h^4}+h^2 }$

8. I'm gonna have to put this one on hold. I emailed my teacher for help. Thanks a lot!

9. Originally Posted by RezMan
I guess not?
so what would I do with $\displaystyle \sqrt{\frac{10000}{h^2}+h^2}$? Should I take the derivative of that or could I reduce it more
This equals
$\displaystyle ( \frac{10000}{h^2}+h^2)^{\frac{1}{2}}$

We take the derivative, first using the power rule, then the chain rule, then... more rules.
The derivative is

$\displaystyle \frac{1}{2} * (\frac{10000}{h^2}+h^2)^{\frac{-1}{2}}$...

times the derivative of
$\displaystyle \frac{10000}{h^2}+h^2}$ ... etc.

10. Originally Posted by RezMan
I have no idea how to solve this
You have the rule for the hypotenuse H in terms of sidelength h. You are expected to find dH/dh using the chain rule, equate to zero etc. etc. Surely you have similar examples in your class notes or textbook.

11. I had a hard time finding the derivative for this. I used a website that said the derivative was this: $\displaystyle \frac{h^4-10000}{h^2\sqrt{h^4+10000}}$

But I only got this far: $\displaystyle \frac{h^4-10000}{h^3\sqrt{\frac{10000+h^4}{h^2}}}$

can someone please tell me if how to get to the one the website gave.

12. Originally Posted by RezMan
I had a hard time finding the derivative for this. I used a website that said the derivative was this: $\displaystyle \frac{h^4-10000}{h^2\sqrt{h^4+10000}}$

But I only got this far: $\displaystyle \frac{h^4-10000}{h^3\sqrt{\frac{10000+h^4}{h^2}}}$

can someone please tell me if how to get to the one the website gave.
In calculus you are expected to be competent with basic algebra. It is essential. Otherwise you are in for a world of pain. It's often been said that most problems students have in calculus are due to poor skills in algebra. I suggest you extensively review all the algebra that is pre-requisite for the subject you are studying.

For the particular problem you're working on, you're meant to note that $\displaystyle \sqrt{\frac{10000 + h^4}{h^2}} = \frac{\sqrt{10000 + h^4}}{h}$ using a basic surd rule .... Now simplify your answer.

13. Sorry about that lol. So:
h^4-10000=(0)h^2sqrt(h^4+10000)
h^4-10000=0
h^4=10000
h=$\displaystyle \pm \sqrt[4]{10000}$
h=10 and it's only positive since it's length we're dealing with right?

So there's only one critical number right? So then I just plug in 10 into this
$\displaystyle \sqrt{\frac{10000}{h^2}+h^2}$ ?