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Simple? Geometry Problem [Pic]

P

PCMarine

Diamond Member
This is an extra credit problem for my pre-calc class:

"A triangle's perimeter is 36. Find the length of the base as to maximize the area."
-Show all functions
-Must use known formulas for all parts.

Pic (Sorry about the crappy MS Paint skills)

Note: The triangle is Isosceles, as legs A and B are congruent. I think to solve this problem, you need two forumlas, which later you substitute into each other to find the answer.

One of the Forumulas is: Base = -2x + 36
The second I think has to do with Area. Area of a triangle is 1/2 B*H. The only problem with using substitution with the 2nd triangle is that you have the height variable, of which, you don't know what the height is.
 
2 unknowns and 2 equations, what's the way it's supposed to be. You do something to both equations to eliminate one of the unknown, then you're golden.
 
Ah, the perimeter is 36. I didn't see that. I was trying to figure out how the hell you were supposed to answer that. lol

Edit:

Nevermind, it's late. 😱
 
Originally posted by: Chaotic42
Yeah, that's what I'm thinking. 12 on each side.
Click to expand...

The problem isn't asking you to change the size of the base so that the area is largest. It's essentially asking "Find the length of the base if the permimeter of the isosceles triangle is 36".

I think IcemanJer is on the right path. However when I plug everything into the formula I get A=-X^2 + 324 which means the area is 18? *shrug*. Anyone want to double check this?
 
The area of the triangle is

A = B*H/2
We need this in terms of B only.

We can use the pythogrean theorem to get H in terms of B and x.

H^2+(B/2)^2=x^2

With the fixed perimeter we can eliminate x.

2X+B=36

H^2+(B/2)^2=((36-B)/2)^2
4H^2=36^2-72B
H=3Sqrt(36-2B)

We now can write the area in terms of B alone.
A=3*B*Sqrt(36-2B)/2
Differentiate wrt B, set =0 and solve for B
dA/dB =3* Sqrt(36-2B)/2 + B(-.5)/sqrt(36-2B)=0

B=12

Edit: opps dropped some constants that disappear in the end
 
Originally posted by: Excelsior
Originally posted by: Chaotic42
Differentiate? I wouldn't think they could use that, since it's not calc.
Click to expand...

You mean IS calc.
Click to expand...

Huh? The guy is in pre-calc, not calculus. I don't think they would have covered differentials in that class, though I could be wrong.
 
This is a very easy problem. Oh, in my pre-calc class we never once used the word derivitive or differentiation. The only thing the teacher came close with was the definition of the derivitive (the point slope formula), but he never called it that. Luckily I taught myself differentiation and integration before that class.
 
Originally posted by: PCMarine
Yea... we haven't covered differentials in my class yet... so my teacher most likely won't accept that 🙁
Click to expand...

If you can't use derivitive, simply plot the equation for Area vs base, pick off the max point.
The derivative gives you the slope of the function, setting it to zero is equivelent to asking where is the equation horizontal. This only occurs at maximum and minimum points. You can do this graphically as well as analyticaly. So for your teacher, follow my development right up to the point were I take a derivitive, instead plot the function for Area. It will have a maximum point at 12.
 
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