Question

a farmer plans to install solar collection panels to provide winter heating for the livestock. the most efficient panel angle for winter heating in the farm’s region is 60° relative to ground level. if an individual panel is 67 inches (in) long and installed on the ground according to these instructions, what is the height, in in, of the upper edge above the ground? 33.5√3 in 33.5√2 in 33.5 in (134√3)/3 in

Explanation:

Step1: Identify the triangle type

We can model this situation as a right triangle, where the length of the solar panel is the hypotenuse (\(c = 67\) in), the angle with the ground is \(\theta=60^{\circ}\), and the height (\(h\)) we need to find is the opposite side to the angle \(\theta\).

Step2: Use the sine function

In a right triangle, the sine of an angle \(\theta\) is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). So, \(\sin(60^{\circ})=\frac{h}{67}\).

We know that \(\sin(60^{\circ})=\frac{\sqrt{3}}{2}\). Substituting this value into the equation:
\(\frac{\sqrt{3}}{2}=\frac{h}{67}\)

Step3: Solve for \(h\)

To find \(h\), we can cross - multiply:
\(h = 67\times\frac{\sqrt{3}}{2}\)
\(67\div2 = 33.5\), so \(h=33.5\sqrt{3}\) in.

Answer:

\(33.5\sqrt{3}\) in