A 130 foot vertical tower is braced with a cable secured to the top of the tower. The angle between the tower and the hill is 90 °. The cable is secured downhill 60 feet from the base of the tower. Calculate the length of the cable and the angle the cable forms with the ground.
sorry. If the tower is vertical, and on a hill, it cannot be perpendicular to the hill. Hills do not have a slope of 0.
To solve this problem, we can use the Pythagorean Theorem and trigonometric ratios.
First, let's label the given information:
The vertical tower has a height of 130 feet.
The cable is secured downhill 60 feet from the base of the tower.
Now, let's find the length of the cable.
Using the Pythagorean Theorem, we can calculate the length of the cable:
(cable length)^2 = (tower height)^2 + (distance downhill)^2
(cable length)^2 = 130^2 + 60^2
(cable length)^2 = 16900 + 3600
(cable length)^2 = 20500
Taking the square root of both sides:
cable length = √20500
cable length ≈ 143.2 feet
Therefore, the length of the cable is approximately 143.2 feet.
Now, let's find the angle the cable forms with the ground.
To do this, we can use trigonometric ratios. Specifically, we can use the sine function.
sin(angle) = (opposite side) / (hypotenuse)
In this case, the opposite side is the tower height (130 feet) and the hypotenuse is the cable length (approximately 143.2 feet).
sin(angle) = 130 / 143.2
Using inverse sine (arcsin) on both sides, we can find the angle:
angle = arcsin(130 / 143.2)
angle ≈ 55.97°
Therefore, the angle the cable forms with the ground is approximately 55.97°.