A vertical tower is supported by guy wires anchored at a point on the ground. The longer wire is attached to the top of the tower and makes an angle of 53 with the ground. The shorter wire makes an angle of 21 with the ground. The shorter wire is 80 ft. long.
Where would the shorter wire be drawn?
a.) how far are the guy wires anchored from the tower?
b.)how long is the longer wire?
c.)how tall is the tower?
To determine where the shorter wire would be drawn, we can use trigonometry and the given information about the angles and lengths of the wires. Here are the steps to find the position of the shorter wire:
1. Draw a diagram representing the tower, the guy wires, and the angles mentioned in the problem. Label the angles and the length of the shorter wire.
2. Let's assume that the base of the tower is located at the origin (0,0) on the coordinate plane. This will make it easier to calculate the position of the shorter wire.
3. Based on the problem, the longer guy wire is attached to the top of the tower and makes an angle of 53 degrees with the ground. This means that we have a right triangle formed with the longer wire as the hypotenuse.
4. Using trigonometry, we can determine the length of the longer wire. The sine function relates the angle opposite the hypotenuse (53 degrees) to the ratio of that side to the hypotenuse. Let's call the length of the longer wire "L":
sin(53) = opposite side (L) / hypotenuse (L)
L = sin(53) * L
5. Now, let's focus on the shorter wire. It is given that the shorter wire makes an angle of 21 degrees with the ground. We have another right triangle formed with the shorter wire as the hypotenuse.
6. Again, using trigonometry, we can find the length of the shorter wire. The sine function relates the angle opposite the hypotenuse (21 degrees) to the ratio of that side to the hypotenuse. Let's call the length of the shorter wire "S":
sin(21) = opposite side (80 ft) / hypotenuse (S)
S = 80 ft / sin(21)
7. With the lengths of the longer and shorter wires determined, we can calculate the position of the shorter wire relative to the base of the tower. Since the base of the tower is at the origin (0,0), the position of the shorter wire would be (0, S), where S represents the calculated length of the shorter wire.
To summarize, the shorter wire would be drawn parallel to the y-axis, starting from the base of the tower at the origin (0,0) and extending upwards by a length of S, which can be calculated using the formula: S = 80 ft / sin(21).