# Thread: Area of a triangle

1. ## Area of a triangle

An isosceles triangle has its vertex at the origin and its base parallel to the x-axis with the vertices above the axis on the curve $y=27-x^2$. Find the largest are the triangle can have.

I ended up with 27, but the answer in the back of the book says 54. I tried redoing the problem but ended up with the same answer. Any help is appreciated, thanks.

2. The triangle's base will have a length $\displaystyle x$ and the height will be $\displaystyle y = 27-x^2$.

So the triangle's area will be $A = \displaystyle \frac{1}{2}x(27-x^2) = \frac{27}{2}x - \frac{1}{2}x^3$.

The maximum will occur where the derivative $\displaystyle = 0$ and the second derivative $\displaystyle < 0$.

So $\displaystyle \frac{dA}{dx} = \frac{27}{2} - \frac{3}{2}x^2=0$

$\displaystyle \frac{27}{2} = \frac{3}{2}x^2$

$\displaystyle 27 = 3x^2$

$\displaystyle 9 = x^2$

$\displaystyle x = \pm 3$.

$\displaystyle \frac{d^2A}{dx^2} = -3x$.

At $\displaystyle x = -3, \frac{d^2A}{dx^2} = 6 > 0$.

At $\displaystyle x = 3, \frac{d^2A}{dx^2} = -6 < 0$.

So the value that has the derivative $\displaystyle =0$ and the second derivative $\displaystyle <0$ is $\displaystyle x=3$.

The area at that point is

$\displaystyle A = \frac{27}{2}(3) - \frac{1}{2}(3)^3$

$\displaystyle = \frac{81}{2} - \frac{27}{2}$

$\displaystyle = \frac{54}{2}$

$\displaystyle = 27$.

I agree with your answer of $\displaystyle 27\,\textrm{units}^2$.

3. I'll have to go with 27 then, but would the triangle being isosceles have something to do with it being 54?

4. Oh, I missed that it was isosceles. If it's an isosceles triangle, then the distance along the base will be doubled.

So the area function is actually $\displaystyle A = \frac{1}{2}(2x)(27-x^2) = 27x - x^3$.

This will affect the final answer.

5. Even after differentiating $27x - x^3$ I still come up with x=3, any idea why?

EDIT: Nevermind, I'm stupid.

6. I think the problem here was trying to understand the question. Drawing out what they were asking usually helps. The triangle they were trying to describe is in the picture below. Once you figured that out, you'll find that Max Area is 54

Originally Posted by lancelot854
Even after differentiating $27x - x^3$ I still come up with x=3, any idea why?

EDIT: Nevermind, I'm stupid.
No you're not stupid; just read the question properly