Question - Solving a Right Triangle Problem with Trigonometry
Solution:
This question involves using trigonometry to solve a right triangle problem. The information provided indicates that Franklin casts a 60° shadow, and the distance from the top of Franklin's head to the shadow's end is 7 feet. The question asks for the length of Franklin's shadow and his height.The 60° angle and the distance from Franklin's head to the shadow's end form a right triangle, with the 7 feet representing the hypotenuse.Let's denote the following:- $$ h $$ as Franklin's height,- $$ s $$ as the length of the shadow,- $$ 60° $$ as the angle of elevation from the end of the shadow to the top of Franklin's head.We can use trigonometry to solve for both $$ s $$ and $$ h $$. More specifically, the cosine and sine functions for a 60° angle in a right triangle.Using the cosine function, which is adjacent over hypotenuse:\[ \cos(60°) = \frac{s}{7} \]Since $$ \cos(60°) = 0.5 $$:\[ 0.5 = \frac{s}{7} \]\[ s = 7 \times 0.5 \]\[ s = 3.5 \text{ feet} \]Using the sine function, which is opposite over hypotenuse:\[ \sin(60°) = \frac{h}{7} \]Since $$ \sin(60°) = \sqrt{3}/2 $$ or approximately 0.8660:\[ 0.8660 = \frac{h}{7} \]\[ h = 7 \times 0.8660 \]\[ h \approx 6.0620 \text{ feet} \]Rounded to one decimal place, Franklin's height would be approximately 6.1 feet.So, the length of Franklin's shadow is 3.5 feet, and his height is approximately 6.1 feet. The correct answer from the options provided would be:A. 3.5 ft., 6.1 ft.
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