A rectangle is inscribed between the x-axis and the parabola y=9-x^2 with one side along the x-axis. One side in touchin the x- axis and the opposite two vertices touch the parabola.
A) Suppose the point (2,5) is the vertex of the rectangle on the original parabola. Use this point to verify your equations from parts a) and b).
I don't really get how I can do that.
Part a) asked for the equation for the area of rectangle- i got -2x^3+18x. Part b) asked for the equation after a horizontal stretch by a factor of 4- I got the y=2x*(-(1/4x)^2+9).
To verify the equations for the rectangle inscribed between the x-axis and the parabola y=9-x^2, we can use the given point (2,5) as the vertex of the rectangle and check if the other vertices satisfy the equations.
Step 1: Find the equation of the parabola:
The given equation of the parabola is y = 9 - x^2.
Step 2: Find the y-coordinate of the other vertex:
Since one side of the rectangle is along the x-axis, the y-coordinate of the other vertex is also 0.
Step 3: Find the x-coordinate of the other vertices:
To find the x-coordinate of the other vertices, we substitute the y-coordinate into the equation of the parabola.
For one vertex: (x, y) = (2, 5)
Substituting into y = 9 - x^2:
5 = 9 - 2^2
5 = 9 - 4
5 = 5
The y-coordinate of the vertex (2,5) satisfies the equation of the parabola.
For the other two vertices: (x, y) = (x, 0)
Substituting into y = 9 - x^2:
0 = 9 - x^2
x^2 = 9
x = ±3
The x-coordinates of the other two vertices are ±3.
Step 4: Verify the equations:
Now we can check if the other two vertices (±3, 0) satisfy the equations:
For the x-coordinate ±3:
Substituting into y = 9 - x^2:
y = 9 - (±3)^2
y = 9 - 9
y = 0
The y-coordinate of the vertices (±3, 0) also satisfy the equation of the parabola.
Therefore, the point (2,5) is verified to be the vertex of the rectangle on the original parabola.
To verify the equations from parts a) and b), we need to check if the given point (2,5) lies on the parabola and satisfies the conditions of the inscribed rectangle.
Let's start by checking if the point (2,5) lies on the parabola y = 9 - x^2.
Substitute x = 2 into the equation of the parabola:
y = 9 - (2^2)
y = 9 - 4
y = 5
As the calculated y-coordinate is 5, which is equal to the given y-coordinate, we can conclude that the point (2,5) lies on the parabola y = 9 - x^2.
Now, let's verify if this point satisfies the conditions of the inscribed rectangle.
The rectangle is inscribed between the x-axis and the parabola, with one side along the x-axis. This means that the bottom side of the rectangle will lie on the x-axis, and the opposite two vertices will touch the parabola.
Since the point (2,5) is given as the vertex of the rectangle on the parabola, we can draw a rectangle with one side along the x-axis, and the other three sides touching the parabola. The bottom side will lie on the x-axis, and the opposite two vertices will be (2,5) and (-2,5) (symmetric to (2,5) with respect to the y-axis).
By drawing this rectangle, you can visually verify if the rectangle satisfies the conditions mentioned.
If the rectangle indeed satisfies these conditions and matches the given point's location, then the equations from parts a) and b) are verified.