Question

error analysis: consider the following problem, which stephanie and adam are both trying to solve:“a cat, who has climbed a tree, looks down at a dog at a 28° angle of depression. if the dog is 34 meters from the base of the tree, how high up is the cat?”the first steps of their work are shown below. analyze their work and determine who, if anyone, has set it up correctly.stephanies work$\tan28^{\circ}=\frac{34}{x}$adams work$\tan28^{\circ}=\frac{x}{34}$determine who set the problem up correctly and determine the height of the cat in the tree.$circ$ stephanies work is correct. the cat is at a height of approximately 64 meters in the tree.$circ$ adams work is correct. the cat is at a height of approximately 64 meters in the tree.$circ$ neither adam or stephanies work is correct. the cat is at a height of approximately 19 meters in the tree.

Explanation:

Step1: Identify angle relationship

The angle of depression equals the angle of elevation from the dog to the cat, so the angle inside the right triangle at the dog is $28^\circ$. The adjacent side to this angle is the 34-meter distance from the dog to the tree, and the opposite side is the height $x$ of the cat.

Step2: Choose correct trigonometric ratio

For the angle $28^\circ$, $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$, so $\tan28^\circ=\frac{x}{34}$. This matches Adam's setup.

Step3: Solve for height $x$

Rearrange to solve for $x$:
$x = 34 \times \tan28^\circ$
Calculate $\tan28^\circ\approx0.5317$, so $x\approx34\times0.5317\approx18.08$, which is approximately 19 meters.

Answer:

Neither Adam or Stephanie's work is correct. The cat is at a height of approximately 19 meters in the tree.