Question

1. a tree growing in a field casts a shadow 40 - foot long. if the angle of elevation to the sun from the shadows tip is 43°. how tall is the tree?

Explanation:

Step1: Set up the trigonometric relationship

We can use the tangent function. Let \(h\) be the height of the tree. The tangent of an angle of elevation \(\theta\) in a right - triangle (where the tree is the opposite side and the shadow is the adjacent side) is given by \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 43^{\circ}\) and the adjacent side \(x = 40\) feet. So, \(\tan(43^{\circ})=\frac{h}{40}\).

Step2: Solve for \(h\)

We know that \(h = 40\times\tan(43^{\circ})\). Using a calculator, \(\tan(43^{\circ})\approx0.9325\). Then \(h=40\times0.9325\).

$$h = 37.3$$

Answer:

The height of the tree is approximately \(37.3\) feet.