Final answer:
The area covered by the windows on the A-frame house can be found using trigonometry by calculating the height of the triangle shaped end using the given roof angle and length, then applying the area formula for a triangle.
Step-by-step explanation:
To calculate the area of the windows on the A-frame house, we consider the end of the house where the windows are to be the shape of an isosceles triangle since the two sides of the roof are equal in length. The peak of the roof creates a 54° angle, which is split into two equal angles at the base, making each base angle 27° (since the total angles in a triangle always add to 180°).
Now we can use trigonometry to find the height (h) of the triangle. Since the triangle is isosceles, we can split it down the middle, creating a right triangle with one angle of 27° and the adjacent side (half the base of the isosceles triangle) unknown. The roof length of 24 feet will be the hypotenuse of the right triangle. Using the cosine function:
cos(27°) = adjacent / hypotenusecos(27°) = (base/2) / 24
Solving for the base (b):
b = 2 * 24 * cos(27°).
Now we can find the height (h) using the sine function:
sin(27°) = opposite / hypotenusesin(27°) = h / 24
Solving for h:
h = 24 * sin(27°).
Finally, we calculate the area (A) of the triangle (which is the area covered by windows):
A = (1/2) * b * h
Inserting our values, we get:
A = (1/2) * 2 * 24 * cos(27°) * 24 * sin(27°)A = 242 * sin(54°) since sin(2x) = 2 * sin(x) * cos(x),
Plugging the values into a calculator gives us the area A, to the nearest hundredth, which is the final answer for the area covered by windows.