Math homework help - word problems?

Question

HELP!!! I can't these word problems.

a) Between 1989 and 1997 the percentage of households with incomes of $100,000 or more is modeled by f(s) .071x squared - .426x+8.05, where x=0 represents the year 1989. Based on this model when did the percent of people in this income level reach its minimum?

b) A rain gutter is to be made up of retangular aluminum sheets 12 inches wide by turning up the sides egdes 90 degree. What depth (of the edges) will provide a maximum cross sectional area and thereby provide for the greatest flow of water?

Answer 1

a) Let f(x) = 0.071x^2-0.426x+8.05
f'(x) = 0.142x-0.426 = 0 set equal to zero to find a max/min
then at x = 0.426/0.142 = 3 there is a min or max ie the slope is zero. So say x = 2, then f'(x) <0 and if x = 4 then f'(x) > 0, so the point is likely a minimum.
Now x=0 is 1989 so that at x=3, the year would be 1989 + 3 = 1992

b) Let x be the height of a side, then 12-2x is the width of the gutter. Then the cross secional area is A(x) = x(12-2x) = 12x-4x^2. A'(x) = 12-8x = 0, set equal to zero to find a max/min.
So A'(x) =0 implies that x = 12/8 or 3/2
A(x=1) = 8
A(x=3/2) = 9
A(x=2) = 8, so that x = 3/2 seems to be the max of area.
Now the height of a side is 3/2 makes a max