Question

1. betty and addie are on either side of a tree that is 20 feet tall. betty sees the top of the tree at 42° and addie sees the top of the tree at 36°. how far apart are betty and addie (to the nearest foot)? show your work #8 15 feet 25 feet 18 feet 50 feet

Explanation:

Step1: Calculate the distance from Betty to the tree

We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For Betty, \(\theta = 42^{\circ}\) and the opposite side (height of the tree) \(h = 20\) feet. Let the distance from Betty to the tree be \(x\). Then \(\tan42^{\circ}=\frac{20}{x}\), so \(x=\frac{20}{\tan42^{\circ}}\).
Using a calculator, \(\tan42^{\circ}\approx0.9004\), so \(x=\frac{20}{0.9004}\approx22.21\) feet.

Step2: Calculate the distance from Addie to the tree

For Addie, \(\theta = 36^{\circ}\) and the opposite side (height of the tree) \(h = 20\) feet. Let the distance from Addie to the tree be \(y\). Then \(\tan36^{\circ}=\frac{20}{y}\), so \(y = \frac{20}{\tan36^{\circ}}\).
Using a calculator, \(\tan36^{\circ}\approx0.7265\), so \(y=\frac{20}{0.7265}\approx27.53\) feet.

Step3: Calculate the total distance between Betty and Addie

The total distance \(d=x + y\). Substitute the values of \(x\) and \(y\): \(d=22.21+27.53 = 49.74\approx50\) feet.

Answer:

50 feet