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 - x². Find the largest area the triangle can have.

Verified step by step guidance

1. Identify the vertices of the isosceles triangle. The vertex is at the origin (0,0), and the base is parallel to the x-axis. The other two vertices are on the curve y = 27 - x^2, so they are at points (x, 27 - x^2) and (-x, 27 - x^2).
2. Express the base length of the triangle. The base of the triangle is the distance between the points (x, 27 - x^2) and (-x, 27 - x^2), which is 2x.
3. Determine the height of the triangle. The height is the y-coordinate of the points on the curve, which is 27 - x^2.
4. Write the formula for the area of the triangle. The area A of a triangle is given by A = 1/2 * base * height. Substitute the expressions for the base and height: A = 1/2 * 2x * (27 - x^2).
5. Maximize the area function. Simplify the area expression to A = x * (27 - x^2) and find the derivative dA/dx. Set the derivative equal to zero to find critical points, and use the second derivative test or analyze the critical points to determine the maximum area.

Key Concepts

Area of a Triangle

The area of a triangle can be calculated using the formula A = 1/2 * base * height. In this context, the base of the isosceles triangle is determined by the x-coordinates of its vertices, while the height is the y-coordinate of the vertex at the origin. Understanding how to express the area in terms of these variables is crucial for maximizing it.

Quadratic Functions

The curve y = 27 - x² is a downward-opening parabola, which is a type of quadratic function. This function's maximum point occurs at its vertex, which is essential for determining the height of the triangle. Recognizing the properties of quadratic functions helps in analyzing the relationship between the triangle's dimensions and the curve.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this problem, we need to maximize the area of the triangle, which requires setting up a function for the area in terms of a variable and then using techniques such as taking derivatives and finding critical points to identify the maximum area.