Question 14 $$ 0 / 5 $$ pts 3 98 Details A sled tavels down a hill. The steepness of the hill varies, making the angle of depression only an apporximate measure of the hill's steepness. If the sled travels 25.2 meters on the hill and the hill is approximately 19.02 meters high, what is the approximate angle of depression? Round your answer to the nearest whole number if necessary. The angle of depression is $$ \approx $$ $$ \square $$ degrees. Submit Question

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To find the approximate angle of depression, we can use the tangent function from trigonometry. The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Given:
- The sled travels 25.2 meters on the hill.
- The hill is approximately 19.02 meters high.

We can use the tangent function to find the angle of depression:
$$ \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} $$

In this case, the opposite side is the height of the hill (19.02 meters) and the adjacent side is the distance the sled travels on the hill (25.2 meters).

$$ \tan(\theta) = \frac{19.02}{25.2} $$

Now, we can calculate the angle of depression using the inverse tangent function:
$$ \theta = \arctan\left(\frac{19.02}{25.2}\right) $$

Let's calculate the angle of depression.
Calculate the value by following steps:
- step0: Calculate:
  $$\arctan\left(\frac{19.02}{25.2}\right)$$
- step1: Divide the terms:
  $$\arctan\left(\frac{317}{420}\right)$$
The approximate angle of depression is approximately 0.646542 radians. To convert this to degrees, we can use the conversion factor where 1 radian is approximately equal to 57.2957795 degrees.

$$ 0.646542 \text{ radians} \approx 0.646542 \times 57.2957795 \text{ degrees} $$

Let's calculate the angle of depression in degrees.
Calculate the value by following steps:
- step0: Calculate:
  $$0.646542\times 57.2957795$$
- step1: Convert the expressions:
  $$\frac{323271}{500000}\times \frac{114591559}{2000000}$$
- step2: Multiply the fractions:
  $$\frac{323271\times 114591559}{500000\times 2000000}$$
- step3: Multiply the terms:
  $$\frac{37044127869489}{500000\times 2000000}$$
- step4: Multiply the terms:
  $$\frac{37044127869489}{1000000000000}$$
The approximate angle of depression is approximately 37.044128 degrees. Rounding to the nearest whole number, the angle of depression is approximately 37 degrees.
The approximate angle of depression is 37 degrees.

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