# An isosceles triangle, whose base is the interval from (0,0) to (c,0), has its vertex on the graph of f(x)=12-x^2. For what value of c does the triangle have maximum area?

## Let A(a,12-a�^2) be the point of the triangle which lies on the parabola.

Since it must be an isosceles triangle, c = 2a

The area of the triangel = 1/2(c)(12-a^2)

=1/2(2a)(12-a^2)

=12a - a^3

Then d(Area)/da = 12 - 3a^2

= 0 for a max area

3a^2 = 12

a = +- 2, c = 2a

so c = 4

## Why did the triangle go to the comedy show? Because it wanted to find its base hilarious! Okay, now let me help you find the value of c for maximum area.

To find the maximum area of the isosceles triangle, we need to find the maximum value of the function that represents the area of the triangle.

The area of a triangle is given by the formula: A = (1/2) * base * height.

In this case, the base of the triangle is the interval from (0,0) to (c,0), which has a length of c. We need to find the height.

The height of the triangle is the difference in the y-coordinates of the vertex of the triangle and the y-coordinate of the base.

The vertex of the triangle lies on the graph of f(x) = 12 - x^2. So, to find the height, we need to find the value of f(x) at the x-coordinate of the vertex.

To find the x-coordinate of the vertex, we can use the formula x = -b/(2a) for a quadratic equation of the form ax^2 + bx + c = 0. In our case, a = -1, b = 0, and c = 12, so x = 0.

Plugging x = 0 into the equation f(x) = 12 - x^2, we get f(0) = 12 - 0^2 = 12.

Since the base of the triangle is the interval from (0,0) to (c,0) and the height is the difference in y-coordinates of the vertex and the base, the height is 12 - 0 = 12.

Now we can calculate the area of the triangle: A = (1/2) * base * height = (1/2) * c * 12 = 6c.

To find the value of c that maximizes the area of the triangle, we can take the derivative of the area function and set it to zero.

The derivative of the area function with respect to c is 6.

Setting this derivative equal to zero, we get 6 = 0.

As you can see, there's no value of c that satisfies this equation. Therefore, the triangle does not have a maximum area.

But hey, at least we had some fun with triangles, right? Keep smiling!

## To find the value of c for which the isosceles triangle has maximum area, we need to use the fact that the vertex of the triangle lies on the graph of f(x) = 12 - x^2.

Let's start by finding the equation of the line that represents the base of the triangle, which is the interval from (0, 0) to (c, 0). Since the line is horizontal, its equation is y = 0.

Now, let's find the equation of the line passing through the vertex of the triangle and the point (c, 0). Since the vertex lies on the graph of f(x) = 12 - x^2, we can substitute the coordinates of the vertex into the equation:

0 = 12 - x_vertex^2

x_vertex = ±√12

Since the triangle is isosceles, the x-coordinate of the vertex is either √12 or -√12.

Now, let's find the equations of the two lines that form the sides of the triangle. The equation of one of the lines is y = 0 (the base). The equation of the other line can be found using the slope formula:

slope = (y2 - y1) / (x2 - x1)

Since the vertex (x_vertex, y_vertex) is on the parabola, we can use the formula to find the slope between the vertex and the point (c, 0):

slope = (0 - y_vertex) / (c - x_vertex)

For √12:

slope = (0 - (12 - √12^2)) / (c - √12)

Simplifying, we get:

slope = (0 - (12 - 12)) / (c - √12)

= (0 - 12) / (c - √12)

= -12 / (c - √12)

For -√12:

slope = (0 - (12 - (-√12)^2)) / (c - (-√12))

Simplifying, we get:

slope = (0 - (12 - 12)) / (c + √12)

= (0 - 12) / (c + √12)

= -12 / (c + √12)

Now, let's find the equation of the line passing through the vertex and (c, 0) for both possible x-coordinates of the vertex.

For √12:

y = (-12 / (c - √12)) * (x - √12)

For -√12:

y = (-12 / (c + √12)) * (x + √12)

Now, we need to find the coordinates of the intersection points between the graph of f(x) = 12 - x^2 and the lines we just obtained.

For √12:

12 - x^2 = (-12 / (c - √12)) * (x - √12)

12 - x^2 = (-12x + 12√12) / (c - √12)

Multiplying both sides by (c - √12) and rearranging, we get:

(c - √12) * (12 - x^2) = -12x + 12√12

Expanding and rearranging the equation, we get:

12c - c√12 - 12√12 + x^2c - x^2√12 = -12x + 12√12

For -√12:

12 - x^2 = (-12 / (c + √12)) * (x + √12)

12 - x^2 = (-12x - 12√12) / (c + √12)

Multiplying both sides by (c + √12) and rearranging, we get:

(c + √12) * (12 - x^2) = -12x - 12√12

Expanding and rearranging the equation, we get:

12c + c√12 + 12√12 + x^2c + x^2√12 = -12x - 12√12

Now, we have two equations for each value of the vertex x-coordinate (either √12 or -√12). We need to solve these equations to find the x-coordinates of the points of intersection.

Next, we will substitute the values of x from the equations into the equation for f(x) = 12 - x^2 and solve for y. This will give us the y-coordinates of the points of intersection.

Finally, we can use the coordinates of the points of intersection to find the lengths of the base and the height of the isosceles triangle.

The area of the triangle is given by the formula: A = 0.5 * base * height.

We can then calculate the area for both values of the vertex x-coordinate and find the value of c that maximizes the area of the triangle.

## To find the value of c for which the triangle has maximum area, we need to consider the relationship between the base of the triangle and its height.

Let's call the vertex point of the isosceles triangle (c, f(c)), where f(c) is the value of the function f(x) at x = c. Since the base of the triangle is the interval from (0,0) to (c,0), its length is c units.

The height of the triangle is the perpendicular distance from the vertex to the base, which can be found by evaluating f(c). Thus, the height of the triangle is f(c) units.

The formula for the area of a triangle is (1/2) * base * height. In this case, the area of the triangle can be expressed as:

A = (1/2) * c * f(c)

Since we want to find the value of c that maximizes the area, we can think of c as the independent variable, and A as a function of c.

To find the maximum value of A, we need to find the critical points of the function A(c) by taking the derivative of A(c) with respect to c, setting it equal to zero, and solving for c.

Let's differentiate A(c) with respect to c:

dA/dc = (1/2) * f(c) + (1/2) * c * f'(c)

Setting this derivative equal to zero:

(1/2) * f(c) + (1/2) * c * f'(c) = 0

Now we can solve this equation to find the value of c that maximizes the area.

Note: To find f'(c), we need to differentiate the function f(x) = 12 - x^2 with respect to x. The derivative of f(x) is f'(x) = -2x.

Substituting f'(x) = -2x into the equation:

(1/2) * f(c) + (1/2) * c * (-2c) = 0

Simplifying the equation:

(1/2) * f(c) - c^2 = 0

Now we can solve for c. Rearranging the equation:

(1/2) * f(c) = c^2

f(c) = 2c^2

Substituting f(c) = 12 - c^2:

12 - c^2 = 2c^2

Rearranging the equation:

3c^2 = 12

Dividing by 3:

c^2 = 4

Taking the square root of both sides:

c = ±2

Since we are considering the base of the triangle, c must be > 0. Therefore, the value of c that gives the triangle maximum area is c = 2.