A telephone pole is secured with two cables. The first cable is attached to the top of the pole and makes an
angle of 80° with the ground. The cable is secured to the ground 7 m from the bottom of the pole. The second
cable is attached to the top of the pole and secured to the ground three times as far from the pole as the first
cable is, on the same side of the pole. Determine the angle the second cable makes with the ground, to the
nearest degree.
Let's start by drawing a diagram to visualize the situation:
We can start by using trigonometry to find the height of the pole. We know that the first cable makes an angle of 80° with the ground, so the angle between the cable and the pole is 10° (since the two angles add up to 90°). We can set up a ratio:
tan 10° = height of pole / distance from bottom of pole to where first cable is attached
tan 10° = h / 7
h = 7 tan 10°
h ≈ 1.23 m
Now we can use the height of the pole and the distance from the second cable to the pole to find the angle between the second cable and the ground. Let's call this angle x. We can set up another ratio:
tan x = height of pole / distance from second cable to pole
tan x = 1.23 / (3 * 7)
tan x ≈ 0.0583
To find the angle x, we need to take the inverse tangent of 0.0583:
x = tan⁻¹ 0.0583
x ≈ 3.33°
Therefore, the second cable makes an angle of approximately 3° with the ground.
To determine the angle the second cable makes with the ground, we can use trigonometry.
Let's break down the problem step-by-step:
Step 1: Draw a diagram representing the situation described in the problem.
We have a telephone pole with two cables. The first cable is attached to the top of the pole and makes an angle of 80° with the ground. The cable is secured to the ground 7 m from the bottom of the pole. The second cable is attached to the top of the pole and secured to the ground three times as far from the pole as the first cable is, on the same side of the pole.
Since we know that the second cable is attached three times as far as the first cable, let's label the distance from the pole to the attachment point of the second cable as 3x.
Step 2: Use trigonometry to find the length of the first cable.
Using trigonometry, we can determine the length of the first cable.
We know that the first cable forms an angle of 80° with the ground and is secured 7 m from the bottom of the pole. We can use the sin function to find the length of the first cable.
sin(80°) = opposite/hypotenuse
sin(80°) = 7/hypotenuse
Rearranging the formula, we get:
hypotenuse = 7/sin(80°)
Use a calculator to find the value of sin(80°) and calculate the length of the first cable.
Step 3: Find the length of the second cable.
We know that the length of the second cable is three times the distance of the first cable from the pole, which we labeled as 3x.
Therefore, the length of the second cable is 3 times the length of the first cable, which is 3 * hypotenuse (from Step 2).
Step 4: Use trigonometry to find the angle the second cable makes with the ground.
We can use trigonometry again to find the angle the second cable makes with the ground.
We know that the second cable forms a right-angled triangle with the ground. Let the angle the second cable makes with the ground be θ.
cos(θ) = adjacent/hypotenuse
cos(θ) = 3x/(3 * hypotenuse)
Rearranging the formula, we get:
θ = cos^(-1)(3x/(3 * hypotenuse))
Substitute the values of x and hypotenuse into the formula to find the value of θ.
Step 5: Round the final answer to the nearest degree.
Round the angle θ obtained in Step 4 to the nearest degree.