From a point on the ground a person notices that a 110-ft tree on top of a hill subtends an angle of 3°. If the angle of elevation to the bottom of the tree, represented by angle A in the figure below, measures 37°, find the height h of the hill

To find the height of the hill, we can use the tangent function. The equation for tangent is:

tan(theta) = opposite / adjacent

In this case, theta represents the angle of elevation to the bottom of the tree, A, which is 37°, and the opposite side is the height of the tree, 110 ft. The adjacent side is the height (h) of the hill.

Therefore, we can set up the following equation:

tan(37°) = 110 ft / h

To solve for h, we can rearrange the equation as follows:

h = 110 ft / tan(37°)

Now we can calculate the value of h by plugging in the numbers:

h = 110 ft / tan(37°)
h ≈ 110 ft / 0.7536
h ≈ 146.03 ft

Therefore, the height of the hill is approximately 146.03 feet.

To find the height h of the hill, we can use trigonometry and the given information about the tree and angles.

Let's consider the right triangle formed by the person, the bottom of the tree, and the top of the hill. In this triangle, the side opposite to angle A is the height of the hill (h), the side adjacent to angle A is the distance from the person to the bottom of the tree (110 ft), and the hypotenuse is the distance from the person to the top of the hill.

Since we have the angle and the side adjacent to it, we can use the tangent function to find the height of the hill:

tan(A) = h / 110

Rearranging the equation, we get:

h = 110 * tan(A)

Now we substitute the given angle A (37°) into the equation:

h = 110 * tan(37°)

Using a calculator, we can find the value of tan(37°) to be approximately 0.7536.

Plugging this value into the equation, we get:

h = 110 * 0.7536

Calculating this, we find:

h ≈ 82.9 ft

Therefore, the height of the hill is approximately 82.9 feet.