A telephone pole is secured with two cables. The first cable is attached to the top of the pole and makes an angle of 70° with the ground. The cable is secured to the ground 8 m from the bottom of the pole. The second cable is attached to the top of the pole and secured to the ground 3.5 times as far from the pole as the first cable is, on the same side of the pole. Determine the length of the second cable, to the nearest tenth of a metre.
cable 2 is attached 3.5*8 = 28m from the pole
pole is of height h = 8tan70° = 22m
cable 2 is sqrt(28^2+22^2) = 35.61m
To determine the length of the second cable, we need to apply trigonometry and use the information given in the problem.
Let's break down the problem into the given information:
- The first cable is attached to the top of the pole and makes an angle of 70° with the ground.
- The first cable is secured to the ground 8 m from the bottom of the pole.
- The second cable is attached to the top of the pole.
- The second cable is secured to the ground 3.5 times as far from the pole as the first cable is, on the same side of the pole.
To find the length of the second cable, we can use the tangent function. The tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side. In this case, the angle is 70°.
Let's use "x" to represent the length of the second cable. We know that the first cable is secured to the ground 3.5 times as far from the pole as the first cable is.
Therefore, the distance from the pole to the securing point of the second cable is 3.5 times 8 m, which is equal to 28 m.
Now, we can set up the equation using the tangent function:
tan(70°) = x / 28
To solve for x, we can multiply both sides of the equation by 28:
28 * tan(70°) = x
Using a calculator, we find that tan(70°) is approximately 2.7475.
So, the length of the second cable, rounded to the nearest tenth of a meter, is:
x = 28 * 2.7475 ≈ 76.9 m
Therefore, the length of the second cable is approximately 76.9 meters.
To determine the length of the second cable, we can use trigonometry. Let's first label the distances and angles given in the problem:
- The distance from the top of the pole to the ground is labeled as "y".
- The angle that the first cable makes with the ground is labeled as "θ" (theta), which is 70°.
- The distance from the first cable's point of attachment to the ground is labeled as "x", which is 8 m.
- The distance from the second cable's point of attachment to the ground is labeled as "3.5x".
Using trigonometry, we can write down the mathematical equation that relates the lengths and angles:
tan(θ) = y / x
To find the length of the second cable, we need to find the value of "y". Rearranging the equation, we get:
y = x * tan(θ)
Now we substitute the values that we know into the equation:
y = 8 * tan(70°)
Using a calculator, we can evaluate this expression:
y ≈ 26.56 m
So the length of the second cable, which is 3.5 times the distance from the first cable's point of attachment to the ground, can be calculated as:
3.5x = 3.5 * 8 = 28 m
Therefore, the length of the second cable is approximately 28 meters to the nearest tenth of a metre.