Explain how to solve the problem below. In your response, you must analyze the given information, discuss a strategy or plan to solve the problem, determine and justify a solution, and evaluate the reasonableness of the solution. Chad casts a shadow that is 14.3 feet long. The straight-line distance from the top of Chad's head to the end of the shadow creates a [tex]$23^{\circ}$[/tex] angle with the ground. How tall is Chad, to the nearest tenth of a foot?
Answer (2)
Use the tangent function to relate the angle of elevation, Chad's height ( h ), and the shadow length ( s ): tan ( θ ) = s h .
Solve for h : h = s ⋅ tan ( θ ) .
Substitute the given values: h = 14.3 ⋅ tan ( 2 3 ∘ ) .
Calculate Chad's height and round to the nearest tenth of a foot: 6.1 feet .
Explanation
Analyze the problem and given data We are given that Chad casts a shadow that is 14.3 feet long, and the angle between the ground and the line from the top of Chad's head to the end of the shadow is 2 3 ∘ . We need to find Chad's height to the nearest tenth of a foot. Let's denote Chad's height as h , the length of the shadow as s , and the angle as θ .
Set up the trigonometric equation We can use the tangent function to relate the angle, the height, and the shadow length. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In this case, the opposite side is Chad's height h , and the adjacent side is the length of the shadow s . So we have: tan ( θ ) = s h
Solve for Chad's height We are given that s = 14.3 feet and θ = 2 3 ∘ . We want to find h . We can rearrange the equation to solve for h :
h = s ⋅ tan ( θ ) Now, we substitute the given values: h = 14.3 ⋅ tan ( 2 3 ∘ )
Calculate the height We know that tan ( 2 3 ∘ ) ≈ 0.42447 . Therefore, h = 14.3 ⋅ 0.42447 ≈ 6.0699
Round to the nearest tenth We need to round the value of h to the nearest tenth of a foot. So, h ≈ 6.1 feet.
State the final answer Therefore, Chad's height is approximately 6.1 feet.
Examples
Understanding trigonometry helps in various real-world scenarios, such as determining the height of buildings or trees using shadows and angles, navigating using GPS, or even in construction to ensure precise measurements and angles for structural integrity.
To find Chad's height, we use the tangent function relating his height to the length of his shadow and the angle of elevation. By applying the formula h = s ⋅ tan ( θ ) , we find that Chad's height is approximately 6.1 feet when rounded to the nearest tenth. This method utilizes basic trigonometry principles to derive our answer.
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