When the angle of elevation of the sun is 61°, a telephone pole that is tilted at an angle of 8° directly away from the sun casts a shadow 20 feet long. Determine the length of the pole to the nearest foot.
I'm stuck halfway through this problem, I know how to find all the angles and side lengths of the triangle made when there isn't the 8 degree tilt, but I don't know how to solve the triangle with the 8 degree tilt. What's the next step in solving the triangle?
I assume you found the vertical height of the pole.
length of pole/verticalheight=cos8
To solve the triangle with the 8-degree tilt, you can use trigonometric ratios. In this case, we can use the tangent function.
Let's label the length of the pole as x.
In the right triangle formed by the pole, the shadow, and the angle of elevation of the sun, the side opposite to the angle of elevation is the length of the pole, x. The side adjacent to the angle of elevation is the shadow length, which is 20 feet.
Using the tangent function, we can set up the following equation:
tan(61°) = x / 20
To solve for x, we multiply both sides of the equation by 20:
20 * tan(61°) = x
Using a calculator, we can find the value of the tangent of 61 degrees and multiply it by 20 to determine the length of the pole.
To solve the triangle with the 8-degree tilt, you can use the concept of complementary angles. Since the telephone pole is tilted 8 degrees directly away from the sun, you can think of it as a right-angled triangle formed by the pole, its shadow, and a line perpendicular to the ground.
Now, let's label the sides of this right-angled triangle. The side opposite the 61-degree angle is the length of the pole, the side opposite the 8-degree angle is the length of the shadow, and the side adjacent to both angles is the vertical tilted side.
From the problem statement, you know that the length of the shadow is 20 feet. You also know that the angle of elevation of the sun is 61 degrees, so the angle between the pole and the ground is its complementary angle, which is 90 - 61 = 29 degrees.
Now, to solve for the length of the pole, you can use the tangent function. Tangent is the ratio of the opposite side (length of the pole) to the adjacent side (length of the shadow).
Using the formula: tan(angle) = opposite/adjacent, you can calculate the length of the pole:
tan(29 degrees) = length of the pole / 20 feet
Rearranging the equation to solve for the length of the pole, you get:
length of the pole = tan(29 degrees) * 20 feet
Using a calculator, evaluate the tangent of 29 degrees and multiply it by 20 feet to find the length of the pole to the nearest foot.
the angle at the base of the pole is
... 90º - 8º
the angle at the end of the shadow is the sun's angle
the 3rd angle is opposite the 20 ft shadow
use the law of sines to find the pole