A rain gutter is to be made of aluminum sheets that are 28 inches wide by turning up the edges 90 degree. See the illustration. (a) What depth will provide maximum cross-sectional area and hence allow the most water to flow? (b) What depths will allow at least 66 square inches of water to flow? (a) The depth that will provide the maximum cross-sectional area is inches. (b) The depths between in. and in. will allow at least 66 square inches of water to flow

(a) The depth that will provide the maximum cross-sectional area is  7 Inches.

(b) (b) The depths between 3 inch and 11 inch will allow at least 66 square inches of water to flow.

Step-By-Step Explanation:

Let the height of the gutter =x

Let the length of the base of the gutter = y

This may illustrate an end view of the gutter: x|_y_|x

The Aluminum Sheet =28 Inches Wide

Therefore: y+2x=28

y=28-2x

(a)Cross Sectional Area of the Gutter= Length X breadth

The Maximum of the area which is a parabola occurs when

A gutter depth of 7 inch will provide maximum cross-sectional area and hence allow the most water to flow.

(b)If the Cross-Sectional Area = 66 square inches

(-2x+6)(x-11)=0

-2x+6=0 OR x-11=0

x=3 or x=11

The depths between 3 inch and 11 inch will allow at least 66 square inches of water to flow.

A) Gutter is 7 inches by 14 inches and depth that will provide max area is 7 inches.

B) Between 3 inches and 11 inches

Step-by-step explanation:

First of all let's define the sides of the gutter.

Let the height of the gutter be x

Also, let the base of the gutter be y

Thus, the total length of the 3 sides will be; x + x + y = 2x + y

Since the total perimeter is 28inches. Thus, 2x + y = 28

And, y = 28 - 2x

Cross sectional area; A = xy

Thus, cross sectional area; A = x(28 - 2x) = 28x - 2x²

So, f(x) = A = -2x² + 28x

Now, x coordinate of vertex is gotten from; -b/2a

Thus, x = - (28/(-2•2)) = 7

Thus,y coordinate at this point is;

y = 28 - 2(7) = 14

Thus maximum area will occur at x = 7 and y = 14.

So maximum area = xy = 14 x 7 = 98 in²

B) We have that;

A = -2x² + 28x

Thus, if we want at least A = 66

Which means

-2x² + 28x = 66

Thus,

-2x² + 28x - 66 = 0

Solving for the roots, we get;

x = 3 inches or 11 inches