Given a triangle with an included angle of $30°$ and a side of length $7$ feet adjacent to the angle, if the area of the triangle is $12.6$ square feet, what is the length of the base adjacent to the angle?

• A. $3.6$ feet
• B. $12.6$ feet
• C. $14$ feet
• D. $7.2$ feet

Answer: (D)

Verified step by step guidance

1. Identify the known elements: the included angle \(30^\circ\), one side adjacent to this angle with length 7 feet, and the area of the triangle as 12.6 square feet.
2. Recall the formula for the area of a triangle when two sides and the included angle are known: \[ \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \] where \(a\) and \(b\) are the sides adjacent to angle \(C\).
3. Assign \(a = 7\) feet (the known side), \(C = 30^\circ\), and let \(b\) be the unknown base length adjacent to the angle that we need to find.
4. Substitute the known values into the area formula: \[ 12.6 = \frac{1}{2} \times 7 \times b \times \sin(30^\circ) \].
5. Solve this equation for \(b\) by isolating it on one side: multiply both sides by 2, divide by \(7 \times \sin(30^\circ)\), and simplify to find the length of the base.