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?
Consider half of the rectangle which lies completely in the first quadrant.
The lower left corner is the origin (0,0), and the upper right corner lies on the curve y=9-x².
The area of the (half) rectangle is therefore:
A(x)
=x*y
=x*(9-x²)
=9x-x³
Differentiate with respect to x,
and equate A'(x) to zero to get the value of x that will result in the maximum area. The area of the required rectangle is twice A(x) because the other half of the rectangle is in the second quadrant.
Do check that A"(x)<0 to confirm that the area is a maximum (and not a minimum).
To find the dimensions of the rectangle with the greatest possible area, we need to understand the problem and come up with a strategy to solve it.
The rectangle's base is on the x-axis, which means the two bottom corners of the rectangle have y-coordinate 0. The upper corners of the rectangle lie on the parabola y=9-x^2.
Let's determine the x-coordinate of these upper corners. Since the base of the rectangle is on the x-axis, the y-coordinate of the upper corners must also be 0. Thus, we can set the equation of the parabola equal to 0 and find the corresponding x-values.
0 = 9 - x^2
Rearranging the equation, we get:
x^2 = 9
Taking the square root of both sides, we have:
x = ±√9
Since we are dealing with a rectangle, we only need to consider the positive values of x. Therefore, x = 3.
Now that we have determined the x-coordinate of the upper corners of the rectangle, we can calculate the y-coordinate using the equation of the parabola:
y = 9 - x^2
y = 9 - 3^2
y = 9 - 9
y = 0
So, the upper corners of the rectangle have coordinates (3,0) and (-3,0).
Now, let's find the area of the rectangle. The area of a rectangle is given by the product of its length and width. In this case, the length of the rectangle is the distance between the x-coordinates of the upper corners, which is 3 - (-3) = 6. The width of the rectangle is the y-coordinate of the upper corners, which is 0.
Therefore, the area of the rectangle is:
Area = length × width
Area = 6 × 0
Area = 0
Hence, the rectangle with the greatest possible area has dimensions of 6 units for length and 0 units for width, resulting in an area of 0.