Answer:
(2/3)√15 wide by 10/3 tall, approx 2.582 by 3.333
Explanation:
You want the dimensions of the largest rectangle that fits under the graph of y = 5 -x² above the x-axis.
Dimensions
The graph of y=5-x² is symmetrical about the y-axis, so the width of it will be x -(-x) = 2x and the height will be 5-x².
Area
The area will be the product of the width and height:
A = WH
A = (2x)(5 -x²) = 10x -2x³
Maximum area
The area will be a maximum where the derivative of the area function is zero.
dA/dx = 10 -6x² = 0
x² = 10/6
x = (√15)/3
The corresponding rectangle dimensions are ...
width = 2x = (2/3)√15
height = 5 -x² = 5 -5/3 = 10/3
The rectangle with largest area is (2/3)√15 wide by 10/3 tall.