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Chapter 5: Problem 42

A 12 -foot solar panel is to be installed on a roof with a \(15^{\circ}\) pitch. Find the length of the vertical brace \(d\) if the panel must be installed to make a \(40^{\circ}\) angle with the horizontal.

Short Answer

Expert verified
The length of the vertical brace \(d\) is equal to the product of 12 and the sine of \(25^{\circ}\). Solve the operation for the final answer.

Step by step solution

01

Identify the Right Triangle

Identify the right triangle that the solar panel, the roof and the vertical support form. The right triangle is a triangle in which one angle measures 90 degrees. Here, the right angle is formed at the intersection between the roof and the vertical support.
02

Calculate the angle

Calculate the interior angle between the roof and the solar panel. This is done by subtracting the pitch of the roof (\(15^{\circ}\)) from the angle of the solar panel with the horizontal direction (\(40^{\circ}\)). So, the angle between the solar panel and the roof would be \(40^{\circ} - 15^{\circ} = 25^{\circ}\).
03

Use the Sine Rule

Using the sine rule in this right triangle, we can express the ratio of the length of the vertical brace to the solar panel as the sine of the angle calculated in Step 2. That is, \(d/12 = \sin(25^{\circ})\).
04

Solve for \(d\)

Rearrange the expression in Step 3 to solve for \(d\). Multiply both sides by 12 to get the length of the vertical brace. Thus, \(d = 12 \times \sin(25^{\circ})\). Calculate \( \sin(25^{\circ})\), multiply by 12 to find the length of \(d\).

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