Alina's cat is stuck in a tree. Alina wants to know how high off the ground her cat is. Alina's eye level is 5.97 feet above the ground, and she is standing 63.7 feet from the base of the tree. The angle of elevation from her eye level is 46 degrees. How high off the ground is her cat? Round your answer to the nearest hundredth if necessary. Height of the cat off the ground $$ \approx \square $$ feet Submit Question

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Let $$ x $$ be the additional height above Alina's eyes to the cat. We have a right triangle where the opposite side is $$ x $$ and the adjacent side is the horizontal distance from the tree, which is $$ 63.7 $$ feet. The tangent of the angle of elevation relates these by

$$
\tan(46^\circ)=\frac{x}{63.7}.
$$

Solving for $$ x $$,

$$
x=63.7 \tan(46^\circ).
$$

Next, the height of the cat off the ground is the sum of Alina's eye level and $$ x $$:

$$
\text{Height of cat} = 5.97 + 63.7 \tan(46^\circ).
$$

Using a calculator, we find

$$
\tan(46^\circ) \approx 1.03553.
$$

Then

$$
x \approx 63.7 \times 1.03553 \approx 65.97.
$$

Thus, the total height is

$$
5.97 + 65.97 \approx 71.94 \text{ feet}.
$$

Height of the cat off the ground $$ \approx 71.94 $$ feet.
The cat is approximately 71.94 feet off the ground.

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