A 20 meter long cable is used to support a telephone pole, holding it perpendicular to the ground. If the cable forms a 60 degree angle with the ground, how high up the pole should the cable be attached?
To find the height at which the cable should be attached to the pole, we can use trigonometry. We know that the cable forms a 60-degree angle with the ground and the cable itself has a length of 20 meters.
Let's assume that the height at which the cable is attached to the pole is h meters.
In a right-angled triangle formed by the pole, the ground, and the cable, the side opposite the 60-degree angle is the height of the pole and the hypotenuse is the cable.
Using the definition of sine, we can write:
sin(60 degrees) = opposite/hypotenuse
sin(60 degrees) = h/20
Since sin(60 degrees) is equal to √3/2,
√3/2 = h/20
To solve for h, we can cross-multiply:
2h = 20 * √3
Divide both sides of the equation by 2:
h = 10√3
Therefore, the cable should be attached approximately 10√3 meters above the ground.
To find out how high up the pole the cable should be attached, we can use trigonometry. Specifically, we can use the sine function since we know the length of the cable and the angle it forms with the ground.
The formula we will use is:
sin(angle) = opposite/hypotenuse
In this case, the angle is 60 degrees and the hypotenuse is the length of the cable, which is 20 meters.
So, let's plug in the values into the formula and solve for the opposite side (the height up the pole):
sin(60 degrees) = height/20 meters
To find the value of sin(60 degrees), we can use a calculator or reference table. The sine of 60 degrees is √3/2.
So, the equation becomes:
√3/2 = height/20 meters
To solve for height, we can cross-multiply:
height = (√3/2) * 20 meters
Calculating this, we get:
height = 20 * √3/2 ≈ 17.32 meters
Therefore, the cable should be attached approximately 17.32 meters up the pole.