Two wires are connected to a tower at the same point on the tower. Wire 1 makes an angle of 45° with the ground and wire 2 makes an angle of 60° with the ground.

b) Which wire is longer? Explain. (2 marks)
c) If the point where the two wires connect to the tower is 35 m above the ground, determine exact expressions for the lengths of the two wires. (4 marks)
d) Determine the length of each wire, to the nearest tenth of a metre. (2 marks)

b) To determine which wire is longer, we need to compare the lengths of the two wires. Let's assume that wire 1 is longer. If wire 1 is longer, then the distance from the point of connection on the tower to the ground is greater compared to wire 2. However, wire 1 makes an angle of 45° with the ground, which is smaller than the angle of 60° that wire 2 makes with the ground.

Since wire 2 makes a larger angle with the ground, it would require a longer length to reach the same point on the ground as wire 1. Therefore, wire 2 is longer than wire 1.

c) Let's represent the length of wire 1 as L1 and the length of wire 2 as L2.

Given that the point where both wires connect to the tower is 35 m above the ground, we can use trigonometry to find the lengths of the wires.

For wire 1 (L1):

We can use the tangent function because the angle between wire 1 and the ground is given as 45°.
tan(45°) = L1 / 35
L1 = 35 * tan(45°)
L1 = 35

For wire 2 (L2):

We can use the tangent function because the angle between wire 2 and the ground is given as 60°.
tan(60°) = L2 / 35
L2 = 35 * tan(60°)
L2 ≈ 35 * 1.732
L2 ≈ 60.62

Therefore, the exact expressions for the lengths of the two wires are:
L1 = 35 m
L2 ≈ 60.62 m

d) To determine the length of each wire to the nearest tenth of a meter:

L1 = 35 m (since the value is already given in meters)
L2 ≈ 60.62 m, rounded to the nearest tenth of a meter, is approximately 60.6 m.

b) To determine which wire is longer, we need to compare the lengths of wire 1 and wire 2.

To do this, we can use trigonometry and the given angles.

Let's assume the length of wire 1 is L1 and the length of wire 2 is L2.

Since wire 1 makes an angle of 45° with the ground, we can use the cosine function to find the length of wire 1:

cos(45°) = adjacent / hypotenuse
cos(45°) = L1 / hypotenuse

Since the adjacent side is the length of wire 1 and the hypotenuse is the distance from the tower to the ground, we can rewrite the equation as:

cos(45°) = L1 / (35 + L1)

Similarly, for wire 2, using the cosine function:

cos(60°) = L2 / (35 + L2)

To determine which wire is longer, we need to compare L1 and L2 by solving the two equations above simultaneously.

c) To find the exact expressions for the lengths of the two wires, we need to solve the equations found in part (b). Let's solve for L1 and L2.

1. From the first equation (cos(45°) = L1 / (35 + L1)):
Simplify by cross-multiplying:
L1 * cos(45°) = 35 + L1

Multiply both sides by √2:
L1 * cos(45°) * √2 = (35 + L1) * √2

Simplify and rearrange:
L1 = (35 + L1) * √2 / √2 * cos(45°)

Simplify further:
L1 = (35 + L1) / cos(45°)

2. From the second equation (cos(60°) = L2 / (35 + L2)):
Simplify by cross-multiplying:
L2 * cos(60°) = 35 + L2

Multiply both sides by 2:
L2 * cos(60°) * 2 = (35 + L2) * 2

Simplify and rearrange:
L2 = (35 + L2) * 2 / cos(60°) * 2

Simplify further:
L2 = (35 + L2) / cos(60°)

d) To determine the length of each wire to the nearest tenth of a metre, substitute the values of L1 and L2 into the equations and evaluate them using a calculator or trigonometric table.

For example, for L1 = (35 + L1) / cos(45°), substitute L1 = 0 and solve for L1. Repeat the same process for L2.

Once you have the exact values of L1 and L2, round them to the nearest tenth of a meter to find the lengths of the wires.

The wire #1 is 35m and wire #2 is 40m