A Norman window has the shape of a rectangle surmounted by a

A Norman window has the shape of a rectangle surmounted by a semi-circle, as shown in the figure below. A Norman window with perimeter 30 ft is to be constructed. Find a function that models the area of the window. A(x) = Find the dimensions of the window that admits the greatest amount of light. (Round your answers to one decimal place.) x = ft height = ft A wire 10 cm long is cut into two pieces, one of length x and the other as shown in the figure. Each piece is bent into the shape of a square. Find a function that models the total area enclosed by the two squares. A(x) = Find the value of x that minimizes the total area of the two squares. x = cm A rain gutter is formed by bending up the sides of a 32-inch wide rectangular metal sheet as shown in the figure. Find a function that models the cross-sectional area of the gutter in terms of x. A(x) = Find the value of x that maximizes the cross-sectional area of the gutter. x = in

Solution

1) As tan 2x = 2*tan x/(1 - tan^2 x)

Here 2*tan 3/(1 - tan^2 3) = tan 6 = 0.105

2) As tan 2x = 2*tan x/(1 - tan^2 x)

Here 2*tan 3/(1 - tan^2 3) = tan 6

3) Please call the height of the rectangle H, and the radius of the semicircle R (making the width of rectangle 2R). The total perimeter is 2H + 2R + R, and the perimeter equals 30 ft:
2H + 2R + R = 30
2H = 30 - 2R - R
H = 15 - R - ½R

The area of the window is A = 2RH + ½R²
Substituting H above, we get:
A = 2R(15 - R - ½R) + ½R²
A = 30R - 2R² - R² + ½R²
A = 30R - 2R² - ½R²
A\' = 30 - 4R - R
0 = 30 - 4R - R
4R + R = 30
R(4+ ) = 30
R = 30/(4+ )
width of rectangle = 2R = 60/(4+) ft. width
height of rectangle = 15 - R - ½R
= 15 - 30/(4+ ) - 15/(4+ )
with a common denominator of 4+ ,
[15(4+ ) - 30 - 15] / (4+)
= 30/(4+) ft. height of rectangle