A straight road makes an angle, A of 12 degrees. When the angle of elevation, B, of the sun is 55 degrees, a vertical pole beside the road casts a shadow 7 feet long parallel to the road. Approximate the length of the pole. Round to tow decimal places.
draw a diagram.
P = end of shadow on road
T = top of pole
Q = foot of pole
Z = intersection of horizontal line from Q and pole extended vertically.
Let d = PT: the distance from the tip of the shadow to the top of the pole
Let x = PZ
Let y = ZQ
Let h = QT, the height of the pole
We know:
∠ZPT = 55°
∠ZPQ = 12°
so, ∠QPT = 43°
PQ = 7
x = 7cos12° = 6.847
y = 7sin12° = 1.455
d = x/cos55° = 7cos12°/cos55° = 11.937
now, we can do this two ways
(1) Pythagorean Theorem
x^2 + (y+h)^2 = d^2
h = 8.323
(2) Law of Cosines
h^2 = d^2 + 7^2 - 2*7*d*cos43°
h = 8.323
To approximate the length of the pole, we can use trigonometry. Let's denote the length of the pole as "x."
From the given information, we can form a right triangle with the pole, its shadow, and the sun's rays. The angle of elevation of the sun, B, is formed by the horizontal road and the sun's rays.
Since the shadow is parallel to the road, this means the opposite side of the right triangle is the shadow's length, which is 7 feet.
Now, we can use the tangent function to relate the angle of elevation and the length of the shadow to the length of the pole:
tan(B) = opposite/adjacent
tan(55°) = 7/x
To solve for x, we can rearrange the equation:
x = 7/tan(55°)
Using a calculator, we can find:
x ≈ 7/1.4281
x ≈ 4.89
Approximately, the length of the pole is 4.89 feet.
To approximate the length of the pole, we can use trigonometry. Let's break down the problem and follow these steps:
Step 1: Understand the problem.
We have a straight road that makes an angle of A = 12 degrees. The angle of elevation of the sun is B = 55 degrees. There is a vertical pole beside the road, and its shadow is 7 feet long parallel to the road. We need to calculate the length of the pole.
Step 2: Draw a diagram.
Draw a vertical pole beside a straight road. Label the angle between the road and the ground as A = 12 degrees and the angle between the sun's rays and the ground as B = 55 degrees. Label the length of the shadow as 7 feet.
Step 3: Apply trigonometry.
Since we have an angle and a side, we can use the tangent function. The tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side. In this case, the tangent of angle B is equal to the length of the pole divided by the length of the shadow.
tan(B) = length of pole / length of shadow
Step 4: Calculate the length of the pole.
Substitute the values into the equation:
tan(55 degrees) = length of pole / 7 feet
To find the length of the pole, we isolate "length of pole" by multiplying both sides of the equation by 7 feet:
length of pole = 7 feet * tan(55 degrees)
Step 5: Use a calculator to find the approximate length of the pole.
Using a calculator, evaluate the expression:
length of pole ≈ 7 feet * tan(55 degrees)
The result rounded to two decimal places will give us the approximate length of the pole.
Note: Make sure your calculator is set to degree mode when entering the angle.