Find the height of a pine tree that casts a 86 -foot shadow on the ground if the angle of elevation of the sun is $29^{\circ} 35^{\prime}$
Answer
Finally, calculating the product, we find that the height of the tree is approximately \(\boxed{48.4}\) feet.
Step 1 ：First, we need to convert the angle of elevation from degrees and minutes to decimal form. We know that 1 degree is equal to 60 minutes, so \(29^\circ 35' = 29 + \frac{35}{60} = 29.5833^\circ\).
Step 2 ：Next, we use the tangent of the angle of elevation, which is the ratio of the height of the tree to the length of the shadow. Let's denote the height of the tree as \(h\). So, we have \(\tan(29.5833^\circ) = \frac{h}{86}\).
Step 3 ：Solving the equation for \(h\), we get \(h = 86 \times \tan(29.5833^\circ)\).
Step 4 ：Using a calculator, we find that \(\tan(29.5833^\circ)\) is approximately 0.5623.
Step 5 ：Substituting this value into the equation, we get \(h = 86 \times 0.5623\).
Step 6 ：Finally, calculating the product, we find that the height of the tree is approximately \(\boxed{48.4}\) feet.
