Find the area and dimensions of the largest rectangle (with sides parallel to the coordinate axes) that can be inscribed i nthe region enclosed by the graphs of f(x)=18-x^2 and g(x)=2x^2-9.
Sounds like you are in Huler's AP calc class :o
To find the area and dimensions of the largest rectangle that can be inscribed in the region enclosed by the graphs of f(x) = 18 - x^2 and g(x) = 2x^2 - 9, we can follow these steps:
Step 1: Identify the region enclosed by the graphs of f(x) and g(x). Start by graphing both functions on the same coordinate system.
Step 2: Find the x-coordinates of the points where f(x) and g(x) intersect. Set the two functions equal to each other and solve for x:
18 - x^2 = 2x^2 - 9
Combine like terms:
3x^2 = 27
Divide both sides by 3:
x^2 = 9
Take the square root of both sides:
x = ±3
So, the x-coordinates of the points of intersection are x = -3 and x = 3.
Step 3: Find the y-coordinates of the points of intersection. Substitute these x-values back into one of the original functions. We'll use f(x) = 18 - x^2:
For x = -3:
f(-3) = 18 - (-3)^2
= 18 - 9
= 9
For x = 3:
f(3) = 18 - 3^2
= 18 -9
= 9
So, the y-coordinates of the points of intersection are y = 9 for both x = -3 and x = 3.
Step 4: Calculate the width and height of the rectangle. The width of the rectangle is the distance between the x-coordinates of the points of intersection, which is 3 - (-3) = 6. The height of the rectangle is the difference between the y-coordinate and the x-axis, which is 9 - 0 = 9.
Step 5: Find the area of the rectangle. The area of a rectangle is given by the formula: Area = width * height. In this case, the area is 6 * 9 = 54 square units.
Therefore, the dimensions of the largest rectangle that can be inscribed in the given region are width = 6 units, height = 9 units, and the area is 54 square units.