Question

homework 4.3
score: 12/15 answered: 14/15
question 6
find ( x ) correct to 2 decimal places.
note: the triangle is not drawn to scale.
(image of a triangle with a height of 82, a 57° angle on the left, a 41° angle on the right, and ( x ) as the base length)
( x = ) input box
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Explanation:

Step1: Find the two segments of the base

Let the left segment (adjacent to \(57^\circ\)) be \(a\) and the right segment (adjacent to \(41^\circ\)) be \(b\). We know the height \(h = 82\).

For the left right triangle (with angle \(57^\circ\)): \(\tan(57^\circ)=\frac{h}{a}\), so \(a=\frac{h}{\tan(57^\circ)}\).

For the right right triangle (with angle \(41^\circ\)): \(\tan(41^\circ)=\frac{h}{b}\), so \(b=\frac{h}{\tan(41^\circ)}\).

Step2: Calculate \(a\) and \(b\)

First, calculate \(a\):
\(\tan(57^\circ)\approx1.5399\), so \(a = \frac{82}{1.5399}\approx53.25\)

Then, calculate \(b\):
\(\tan(41^\circ)\approx0.8693\), so \(b=\frac{82}{0.8693}\approx94.33\)

Step3: Calculate \(x\)

\(x=a + b\), so \(x\approx53.25+94.33 = 147.58\)

Answer:

\(147.58\)