A pole leans away from the sun at an angle of 7° to the vertical. When the angle of elevation of the sun is 51°, the pole cast a shadow 47ft. long on level ground. How long is the pole?
using the law of sines, the pole's height h can be found after determining all the angles involved.
h/sin51° = 47/sin47°
To solve this problem, we can use the concept of similar triangles. Let's denote the length of the pole as 'x'.
Step 1: Draw a diagram
Draw a vertical line to represent the pole. From the top of the pole, extend a line to represent the shadow cast on the ground. Label the length of this line as 47ft.
Step 2: Identify the relevant angles
The angle of elevation of the sun is given as 51°. The angle between the pole and the vertical is given as 7°.
Step 3: Identify the similar triangles
In the diagram, we can see that we have two triangles: the vertical line (pole) and the shadow line on the ground.
Step 4: Set up the ratios
Using the ratios of corresponding sides of similar triangles, we can set up the following equation:
tan(51°) = opposite/adjacent
tan(7°) = opposite/47ft
Step 5: Solve for the length of the pole
Using the equations from step 4, we can solve for 'x':
tan(51°) = x/47ft
tan(7°) = x/47ft
x = tan(7°) * 47ft / tan(51°)
Step 6: Calculate the length of the pole
Using a calculator, compute the value of x:
x ≈ (0.122784) * 47ft / (0.777146)
x ≈ 7.386 ft
Therefore, the length of the pole is approximately 7.386 feet.
To find the length of the pole, we can use the concept of trigonometry. Let's call the length of the pole "x".
Given that the pole leans away from the sun at an angle of 7° to the vertical, we can consider the pole as the hypotenuse of a right triangle. The base of the triangle is the shadow cast on the ground, which is 47 feet. The angle of elevation of the sun, which is the angle between the sun's rays and the horizontal ground, is 51°.
Now, we can use the tangent function to relate the angle of elevation with the length of the pole. The tangent of an angle is equal to the ratio of the opposite side (in this case, the length of the pole) to the adjacent side (in this case, the length of the shadow).
So, we can set up the equation: tan(51°) = x / 47.
To solve for x, we can rearrange the equation: x = tan(51°) * 47.
Calculating the value of tan(51°) yields approximately 1.313.
Therefore, the length of the pole, x, is approximately 1.313 * 47 = 61.711 feet.
Hence, the length of the pole is approximately 61.711 feet.