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, and the shadow length: tan ( 2 3 ∘ ) = 14.3 h .
Solve for Chad's height by multiplying both sides by 14.3: h = 14.3 × tan ( 2 3 ∘ ) .
Calculate the value of tan ( 2 3 ∘ ) and then multiply by 14.3: h ≈ 14.3 × 0.42447 ≈ 6.0699 .
Round the result to the nearest tenth of a foot: 6.1 feet .
Explanation
Analyze the problem Let's analyze the given information. We know the length of Chad's shadow is 14.3 feet, and the angle of elevation from the end of the shadow to the top of Chad's head is 2 3 ∘ . We need to find Chad's height.
Set up the trigonometric equation We can use the tangent function to relate the angle of elevation, the length of the shadow, and Chad's height. The tangent of an angle in a right triangle is the ratio of the opposite side (Chad's height) to the adjacent side (the length of the shadow). So, we have: tan ( θ ) = adjacent opposite = shadow length height In our case, θ = 2 3 ∘ , the shadow length is 14.3 feet, and we want to find the height h .
Solve for Chad's height Now, we can write the equation: tan ( 2 3 ∘ ) = 14.3 h To solve for h , we multiply both sides of the equation by 14.3: h = 14.3 × tan ( 2 3 ∘ )
Calculate the height Using a calculator, we find that tan ( 2 3 ∘ ) ≈ 0.42447 . Therefore, h = 14.3 × 0.42447 ≈ 6.0699
State the final answer Rounding to the nearest tenth of a foot, we get: h ≈ 6.1 feet So, Chad is approximately 6.1 feet tall.
Examples
Understanding angles of elevation and trigonometric relationships is useful in many real-world scenarios. For example, surveyors use these principles to determine the heights of buildings or mountains by measuring the angle of elevation from a known distance. Similarly, sailors and pilots use angles of elevation and depression to navigate and determine distances to landmarks or other vessels. These calculations are essential in fields like construction, navigation, and astronomy.
To find Chad's height using his shadow length and the angle of elevation, we use the tangent function. Solving the equation gives us Chad's height of approximately 6.1 feet. This method effectively demonstrates how trigonometry can be applied to solve real-world problems.
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