A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 18° with the horizontal. The flagpole casts a 14-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 21°.
(b) Write an equation involving the unknown quantity h.
(c) Find the height of the flagpole. (Round your answer to two decimal places.)
h = m
define
point A is flagpole base
Point B is flagpole top
point C is where shadow ends up slope
<ABC = 90-21 = 69
<BAC = 90-18 = 72
so
<BCA = 180 - 69 - 72 = 39
now law of sines
sin 69 / 14 = sin 39 / h
To solve this problem, we can use trigonometry. We'll break it down step by step:
Step 1: Identify the given information:
- The angle of the slope with the horizontal is 18°.
- The angle of elevation from the tip of the shadow to the sun is 21°.
- The length of the shadow cast by the flagpole is 14 meters.
Step 2: Draw a diagram:
First, draw a line for the slope, making an angle of 18° with the horizontal. At the bottom of the slope, draw a line perpendicular to the slope, representing the flagpole. Then, draw a line from the tip of the shadow to the top of the flagpole, making an angle of 21° with the horizontal. Label the length of the shadow as 14 meters, and let h represent the height of the flagpole.
Step 3: Write the equation:
Using the given angles, we can create a right-angled triangle with the base as the shadow (14 meters), the height as the height of the flagpole (h), and the hypotenuse as the line from the tip of the shadow to the top of the flagpole. By using trigonometry, we can deduce the following equation:
tan(21°) = h/14
Step 4: Solve the equation for h:
To find the value of h, we can rearrange the equation and solve for h:
h = 14 * tan(21°)
Using a calculator, we evaluate the expression to find h. Round the result to two decimal places:
h ≈ 5.65 meters
So, the height of the flagpole is approximately 5.65 meters.