Question

prompt
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 chads head to the end of the shadow creates a 23° angle with the ground. how tall is chad, to the nearest tenth of a foot?

Explanation:

Step1: Identify the right - triangle relationship

We have a right - triangle where the length of the shadow is the adjacent side to the given angle and Chad's height is the opposite side. Let Chad's height be $h$. The adjacent side length $a = 14.3$ feet and the angle $\theta=23^{\circ}$. We use the tangent function $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.

Step2: Set up the equation

The formula for tangent is $\tan\theta=\frac{h}{a}$. Substituting $\theta = 23^{\circ}$ and $a = 14.3$ into the formula, we get $\tan(23^{\circ})=\frac{h}{14.3}$.

Step3: Solve for $h$

We can rewrite the equation as $h = 14.3\times\tan(23^{\circ})$. Since $\tan(23^{\circ})\approx0.4245$, then $h=14.3\times0.4245$.
$h = 14.3\times0.4245=6.07035\approx6.1$ feet.

Step4: Justify the solution

We used the correct trigonometric ratio for a right - triangle. The tangent of an angle in a right - triangle relates the opposite and adjacent sides. Given the information about the adjacent side and the angle, this is the appropriate method.

Step5: Evaluate reasonableness

A height of 6.1 feet is a reasonable height for a person. It is within the typical range of human heights.

Answer:

$6.1$ feet