Answer:
- a) max area for depth of 3 inches
- b) ≥ 16 in² for 2 in ≤ depth ≤ 4 in
Step-by-step explanation:
(a) For a depth of x, the two sides of the rain gutter are length x, and the bottom is length (12-2x). The cross sectional area is the product of these dimensions:
A = x(12 -2x)
This equation describes a parabola that opens downward. It has zeros at ...
x = 0
12 -2x = 0 . . . . x = 6
The maximum area is halfway between these zeros, at x=3.
The maximum area is obtained when the depth is 3 inches.
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(b) For an area of at least 16 square inches, we want ...
x(12 -2x) ≥ 16
x(6 -x) ≥ 8 . . . . . divide by 2
0 ≥ x² -6x +8 . . . . subtract the left side
(x -4)(x -2) ≤ 0 . . . factor
The expression on the left will be negative for values of x between 2 and 4 (making only the x-4 factor be negative). Hence the the depths of interest are in that range.
At least 16 square inches of water will flow for depths between 2 and 4 inches, inclusive.
