A rectangle with its base on the x axis is to be inscribed under the positive portions of the graph of y=8-x^2 . Write an equation for the area of the rectangle which uses only the variable x.
y is positive from x = -sqrt8 to +sqrt8
The area is Integral (8-x^2) dx
from x = -sqrt8 to +sqrt8
eh? We're not looking for the area under the parabola. We want the area of a rectangle inscribed under it.
Let the width of the rectangle be 2x. That is, x is the distance from (0,0) to a bottom corner of the rectangle.
the area is thus
f(x) = 2xy = 2x(8-x^2)
The domain of f is obviously [-√8,√8]
To find the area of the rectangle, we need to determine the length of its base and height.
Since the base of the rectangle is on the x-axis, it means the top side of the rectangle lies on the curve of the equation y = 8 - x^2. To find the points of intersection, we set y = 0 and solve for x:
0 = 8 - x^2
Rearranging the equation, we get:
x^2 = 8
Taking the square root of both sides, we have:
x = ±√8
Since we only want the positive portion of the graph, we will consider x = √8 as the value for half the length of the base of the rectangle.
Thus, the length of the base is 2 times the value of x:
base = 2x = 2√8
Now, to find the height of the rectangle, we need to determine the corresponding y-coordinate on the curve. Let's substitute the value of x back into the equation y = 8 - x^2:
y = 8 - (√8)^2
y = 8 - 8
y = 0
Therefore, the height of the rectangle is the value of y when x = √8, which is 0.
Finally, the equation for the area of the rectangle is given by:
Area = base × height = (2√8) × 0 = 0
Hence, the equation for the area of the rectangle in terms of x is Area = 0.