A solar panel a = 12 feet in width, which is to be attached to a roof that makes an angle of 25° with the horizontal. The bottom edge of the solar panel makes contact with the roof. The top edge of the solar panel is elevated above the roof by a vertical brace of length d. Approximate the length d (in feet) of the vertical brace that is needed for the panel to make an angle of 45° with the horizontal.

Question

A solar panel a = 12 feet in width, which is to be attached to a roof that makes an angle of 25° with the horizontal. The bottom edge of the solar panel makes contact with the roof. The top edge of the solar panel is elevated above the roof by a vertical brace of length d. Approximate the length d (in feet) of the vertical brace that is needed for the panel to make an angle of 45° with the horizontal.

Answer 1

angle D = 45 - 25 = 20 degrees

call point B at the very top of the panel
then the angle at B = 180 - 90 - 45 = 45 degrees
then the last angle between the roof and your holding pole is
180 -45 - 20 = 115 degrees
then
sin 20 / d = sin 115 /12

Answer 2

To approximate the length d of the vertical brace needed for the solar panel to make an angle of 45° with the horizontal, we can use trigonometry.

Let's break down the problem step by step:

1. Find the height of the solar panel (h):
Since the panel is attached to the roof, the height of the solar panel will be the same as the length of the vertical brace (d).

2. Find the vertical displacement (y) of the top edge of the panel when it is at an angle of 45° with the horizontal:
The vertical displacement can be calculated using the formula: y = h * tan(45°).

3. Find the horizontal displacement (x) of the top edge of the panel when it is at an angle of 45° with the horizontal:
The horizontal displacement can be calculated using the formula: x = h * tan(45°).

4. Find the distance (D) between the bottom and the top edge of the solar panel when it is at an angle of 45° with the horizontal:
The distance can be calculated using the Pythagorean theorem: D = sqrt(x^2 + y^2).

Now, let's calculate the values step by step:

Step 1:
Since the height of the solar panel is the same as the length of the vertical brace, h = d.

Step 2:
y = h * tan(45°)
= d * tan(45°)

Step 3:
x = h * tan(45°)
= d * tan(45°)

Step 4:
D = sqrt(x^2 + y^2)
= sqrt((d * tan(45°))^2 + (d * tan(45°))^2)
= sqrt(2 * d^2 * tan^2(45°))
= sqrt(2 * d^2)

To find the approximate length d, we can solve the equation D = 12 (as given in the problem).

sqrt(2 * d^2) = 12
2 * d^2 = 12^2
2 * d^2 = 144
d^2 = 72
d ≈ sqrt(72)
d ≈ 8.49 feet

Therefore, the approximate length d of the vertical brace needed for the panel to make an angle of 45° with the horizontal is approximately 8.49 feet.