Two pieces of wire are stretched from the top M, of a vertical pole to the points L and N on the horizontal ground, The angle of depression of L from M is 65 degrees. N is on the opposite side of the pole (from L) at an angle of depression of 72 degrees from M. N is 4m from the bottom P of the pole.

Question

Two pieces of wire are stretched from the top M, of a vertical pole to the points L and N on the horizontal ground, The angle of depression of L from M is 65 degrees. N is on the opposite side of the pole (from L) at an angle of depression of 72 degrees from M. N is 4m from the bottom P of the pole.

Find the length of each piece of wire

Answer 1

cos72 = NP/MN = 4/MN

MN = 4/cos72 = 12.94 m. = Length of wire to point N.

tan72 = MP/NP = MP/4
MP = 4*tan72 = 12.31 m.

sin65 = MP/LM = 12.31/LM
LM = 12.31/sin65 = 13.58 m. = Length of
wire to point L.

Answer 2

To find the length of each piece of wire, we can use trigonometry. Let's label the length of wire from M to L as 'x' and the length of wire from M to N as 'y'.

First, let's draw a diagram to visualize the situation:

```
M
/|
x / |
/ |y
/ |
/ |
P/_____|\
L N
```

From the given information, we know that the angle of depression of L from M is 65 degrees and the angle of depression of N from M is 72 degrees. We also know that N is 4m from the bottom P of the pole.

Using the trigonometric property for the angle of depression, we can set up the following equations:

tan(65°) = x / PM

tan(72°) = y / PM

To solve for x and y, we need to find PM, which is the vertical height from M to P.

Since N is 4m from P, we can calculate PM as follows:

PM = PN + NM
= 4m + y

Now, we substitute PM into our equations:

tan(65°) = x / (4m + y)

tan(72°) = y / (4m + y)

Next, we can solve these equations simultaneously to find the values of x and y.

First, rearrange the equations to solve for x:

x = tan(65°) * (4m + y)

Now, substitute this expression for x into the second equation:

tan(72°) = y / (4m + y)

tan(72°) = (tan(65°) * (4m + y)) / (4m + y)

Now, solve this equation for y by rearranging and isolating y:

y = (tan(72°) * (4m + y)) / tan(65°) - y

Finally, we can solve this equation for y using algebraic methods such as factoring, expanding, and isolating y.

Once y is found, we substitute its value back into the equation for x to calculate x.

Now, we have the values of x and y, which represent the lengths of the wires from M to L and M to N, respectively.