A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=9–x2. What are the dimensions of such a rectangle with the greatest possible area?
Please show me how you tried to do all these problems. I already know how. There is no point in me doing them all.
To find the dimensions of the rectangle with the greatest possible area, we need to analyze the given information and identify the formula for the area of a rectangle.
Let's start by visualizing the problem. A rectangle is inscribed in the area under the parabola y = 9 - x^2, with its base on the x-axis. Since the base lies on the x-axis, the coordinates of the rectangle's vertices will be (x1, 0), (x2, 0), (x1, y), and (x2, y), where y = 9 - x^2.
To calculate the area of the rectangle, we need to find the length and width. The length of the rectangle will be the difference between the x-coordinates of the upper corners: x2 - x1. The width of the rectangle will be y, which is given by y = 9 - x^2.
Therefore, the area (A) of the rectangle is given by A = (x2 - x1) * y.
To find the dimensions (x1 and x2) that maximize the area, we need to find the maximum value of the area (A).
Let's find the derivative of A with respect to x2 and x1, set it equal to 0, and solve for x1 and x2 to get the critical points.
dA/dx1 = -y (derivative of A with respect to x1)
dA/dx2 = y (derivative of A with respect to x2)
First, let's find y:
y = 9 - x^2
Now, let's find the derivative of A with respect to x1:
dA/dx1 = 0 - (x2 - x1) * 2x1
= -(x2 - x1) * 2x1
And the derivative of A with respect to x2:
dA/dx2 = (x2 - x1) * 2x2
Setting dA/dx1 = 0:
-(x2 - x1) * 2x1 = 0
Simplifying:
(x2 - x1) * x1 = 0
This equation has two possibilities:
1) (x2 - x1) = 0
In this case, x1 = x2, which means the rectangle is actually a square. However, since the base of the rectangle is on the x-axis, the square would also need to have one side on the x-axis. This is only possible when x1 = x2 = 0, which does not generate a rectangle within the given conditions.
2) x1 = 0
If x1 = 0, then the right vertical side of the rectangle aligns with the y-axis. Therefore, the rectangle has its maximum area when x1 = 0.
Next, let's set dA/dx2 = 0:
(x2 - x1) * 2x2 = 0
This equation has two possibilities:
1) (x2 - x1) = 0
As mentioned before, x1 = x2 = 0 does not generate a rectangle within the given conditions.
2) 2x2 = 0
If 2x2 = 0, then x2 = 0.
So, we have found that one of the critical points is x1 = 0 and x2 = 0.
Now let's evaluate the area (A) at the critical points:
A = (x2 - x1) * y
= x2 * (9 - x^2)
At x1 = 0 and x2 = 0, we have:
A = 0 * (9 - 0^2)
= 0
Therefore, the area at this critical point is A = 0.
Since we have only found one critical point and it yields an area of 0, it means no rectangle within the given conditions can have a greater area. Therefore, there is no rectangle with the greatest possible area as specified in the question.