Question

in one area, the lowest angle of elevation of the sun in winter is 24° 22. find the minimum distance, x, that a plant needing full sun can be placed from a fence 4.18 ft high. the minimum distance is
(type an integer or a decimal rounded to the nearest tenth as needed.)

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the opposite side to the angle \(24^\circ22'\) is \(4.18\) ft and the adjacent side is \(x\) (the distance we need to find). So we use the tangent function, which is \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). First, convert \(24^\circ22'\) to decimal degrees. \(22'\) is \(\frac{22}{60}\approx0.3667^\circ\), so \(24^\circ22'\approx24.3667^\circ\).

Step2: Set up the tangent equation

\(\tan(24.3667^\circ)=\frac{4.18}{x}\). We can solve for \(x\) by rearranging the formula: \(x = \frac{4.18}{\tan(24.3667^\circ)}\).

Step3: Calculate the value

First, find \(\tan(24.3667^\circ)\approx\tan(24 + \frac{22}{60})\approx\tan(24.3667)\approx0.453\) (using a calculator). Then \(x=\frac{4.18}{0.453}\approx9.2\) (rounded to the nearest tenth).

Answer:

9.2