two pieces of wire 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 N =65 degrees. N is on the opposite side of the pole(from left)at an angle of depression of 72 degrees from M.
Find the length of each piece of wire.
Not enough information the way you typed it.
Was the height of the pole given?
or was the distance between L and N given ?
To solve this problem, we can use trigonometry and create a diagram to visualize the situation.
Let's label the length from the top of the pole to point L as "x" and the length from the top of the pole to point N as "y".
From the given information, we know that the angle of depression from L to N is 65 degrees and the angle of depression from N to M is 72 degrees.
Now, let's break down the problem into two parts:
1. Finding the length of the wire from the top of the pole to point L (x):
In the right triangle formed by the vertical pole, the horizontal ground, and the wire to point L, the angle opposite to the wire (angle M) is 90 degrees.
Using trigonometry (tangent function), we can determine the length x:
tan(65 degrees) = x / M
x = M * tan(65 degrees)
2. Finding the length of the wire from the top of the pole to point N (y):
In the right triangle formed by the vertical pole, the horizontal ground, and the wire to point N, the angle opposite to the wire (angle M) is 90 degrees.
Using trigonometry (tangent function), we can determine the length y:
tan(72 degrees) = y / M
y = M * tan(72 degrees)
Now we have expressions for both x and y. To find their values, we need to know the value of M.
If you provide the value of M, we can substitute it into the equations to find the lengths x and y.
To find the length of each piece of wire, we can use trigonometry.
Let's assign variables to the lengths of the two wires. Let the length of LM be x, and the length of MN be y.
First, we need to find the length of LN, which represents the distance between L and N on the ground. We can use the angle of depression of L from N, which is 65 degrees.
Using trigonometry, we can use the tangent function:
tan(65) = LN / x
Rearranging the equation, we can solve for LN:
LN = x * tan(65)
Next, let's find the length of MN using the angle of depression of N from M, which is 72 degrees.
Using trigonometry again with the tangent function:
tan(72) = LN / y
Rearranging the equation and substituting the value of LN, we can solve for y:
y = LN / tan(72)
Now, substitute the value of LN from the earlier equation:
y = (x * tan(65)) / tan(72)
Now we have expressions for LN and MN in terms of x. To find the length of each piece of wire, we need to determine the sum of these lengths:
Total length = LM + MN
Substituting the values:
Total length = x + (x * tan(65)) / tan(72)
Simplifying the equation further:
Total length = x * (1 + tan(65) / tan(72))
Now, we have an equation to find the total length in terms of x. To find the length of each piece of wire, we need to solve this equation. You can substitute the values of tan(65) and tan(72) into the equation and then use a calculator to find the solution.