Question

in one area, the lowest angle of elevation of the sun in winter is 24° 15. find the minimum distance x that a plant needing full - sun can be placed from a fence 4.78 ft high. the minimum distance is (type an integer or a decimal rounded to the nearest tenth as needed.)

Explanation:

Step1: Convert angle to decimal degrees

First, convert $24^{\circ}15'$ to decimal - degrees. Since $1^{\circ}=60'$, then $15'=\frac{15}{60}=0.25^{\circ}$. So the angle $\theta = 24.25^{\circ}$.

Step2: Use tangent function

We know that in a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, the height of the fence (opposite side) is $h = 4.78$ ft and the distance we want to find is $x$ (adjacent side). So $\tan\theta=\tan(24.25^{\circ})=\frac{4.78}{x}$.

Step3: Solve for $x$

Rearranging the formula $x=\frac{4.78}{\tan(24.25^{\circ})}$.
We know that $\tan(24.25^{\circ})\approx0.451$. Then $x=\frac{4.78}{0.451}\approx10.6$.

Answer:

$10.6$