Question

A rectangle has its base on the x-axis and its upper two vertices on the parabola y=64−x

2
. What is the largest possible area (in squared units) of the rectangle? squared units

Answer 1

The largest possible area of the rectangle is 0 squared units.

Step-by-step explanation:

To find the largest possible area of the rectangle, we need to find the maximum y-coordinate on the parabola y = 64 - x^2. To do this, we need to find the x-coordinate at the vertex of the parabola. The x-coordinate of the vertex is given by the formula x = -b/2a, where the equation of the parabola is in the form y = ax^2 + bx + c. In this case, the equation of the parabola is y = -x^2 + 64. So, the x-coordinate of the vertex is x = -0/2(-1) = 0.

Substituting x = 0 into the equation of the parabola, we get y = 64 - 0^2 = 64. Therefore, the largest possible area of the rectangle is given by the equation A = 2x(64 - x^2) = 2(0)(64) = 0 squared units.

Answer 2

The largest possible area of the rectangle is 0 square units.

Step-by-step explanation:

To find the largest possible area of the rectangle, we need to find the x-coordinate of the upper two vertices of the rectangle on the parabola y = 64 - x^2.

Since the base of the rectangle is on the x-axis, it means the y-coordinate of the two vertices is 0. Substituting y = 0 into the equation of the parabola, we get 0 = 64 - x^2. Rearranging, x^2 = 64.

Taking the square root of both sides, we get x = ±8. So, the two possible x-coordinates of the vertices are -8 and 8.

Now, we can find the length of the base of the rectangle by subtracting the x-coordinates: 8 - (-8) = 16. Since the height of the rectangle is the y-coordinate of the vertices, which is 0, the area of the rectangle is simply the product of the base and the height: 16 * 0 = 0. Therefore, the largest possible area of the rectangle is 0 square units.