Two wires support an electricity pole as shown in Fig. 5.17.if the wires make angles of 58 and 67 with the ground and /AB/= 21m calculate the lengths of the wires
no diagram. But if you remember your basic trig functions, then if the height of the pole is h, and the wires are on opposite sides of the pole, then
h cot58° + h cot 67° = 21
h = 20
So the wire lengths are 23.6 and 21.7 meters
To solve this problem, we can use the trigonometric ratios sine, cosine, and tangent. Let's call the length of the wire attached to point A as "x" and the length of the wire attached to point B as "y".
Since the wires make an angle with the ground, we can use the sine function to find the vertical heights of each wire. The sine of an angle is equal to the opposite side over the hypotenuse.
For wire AB:
sin(58°) = height of AB / 21m
height of AB = sin(58°) * 21m
For wire BC:
sin(67°) = height of BC / 21m
height of BC = sin(67°) * 21m
Now, we need to find the horizontal distances for each wire. We can use the cosine function which is equal to the adjacent side over the hypotenuse.
For wire AB:
cos(58°) = horizontal distance of AB / 21m
horizontal distance of AB = cos(58°) * 21m
For wire BC:
cos(67°) = horizontal distance of BC / 21m
horizontal distance of BC = cos(67°) * 21m
Let's calculate the values:
For wire AB:
height of AB = sin(58°) * 21m
≈ 0.848 * 21m
≈ 17.808m
horizontal distance of AB = cos(58°) * 21m
≈ 0.534 * 21m
≈ 11.214m
For wire BC:
height of BC = sin(67°) * 21m
≈ 0.921 * 21m
≈ 19.341m
horizontal distance of BC = cos(67°) * 21m
≈ 0.389 * 21m
≈ 8.179m
So, the length of the wire attached to point A (AB) is approximately 17.808 meters, and the length of the wire attached to point B (BC) is approximately 19.341 meters.
To calculate the lengths of the wires, we can use trigonometric ratios. Let's call the length of the wire on the left side of the electricity pole "x" and the length of the wire on the right side "y".
From the given information, we can see that the angles made by the wires with the ground are 58° and 67°. We also know that the horizontal distance AB between the wires is 21m.
To calculate the lengths of the wires, we will use the tangent trigonometric ratio, which is defined as the ratio of the opposite side to the adjacent side of a right triangle.
For the left wire (x):
tan(58°) = x / AB
Rearranging the equation, we get:
x = AB * tan(58°)
For the right wire (y):
tan(67°) = y / AB
Rearranging the equation, we get:
y = AB * tan(67°)
Now, we can substitute the given values and calculate the lengths of the wires:
x = 21 * tan(58°)
y = 21 * tan(67°)
Using a scientific calculator, we can calculate the values of x and y:
x ≈ 34.02 m
y ≈ 47.98 m
Therefore, the length of the wire on the left side is approximately 34.02 meters, and the length of the wire on the right side is approximately 47.98 meters.