A wire to the top of a tower makes an angle of 52 with the le el ground. At a point 32 ft. Farther from the base of the toser and in líne with the base of the wire, the angle of elevation to the top of the toser is 21. What is the length of the tower.
draw a diagram, as usual. If the tower has height h and the wire is meets the ground at distance d,
h/d = sin 52
h/(d+32) = sin 21
eliminate d and you have
h/sin52 = h/sin21 - 32
now just solve for h
To find the length of the tower, we can use trigonometry. Let's break down the problem step by step:
Step 1: Draw a diagram
Draw a diagram of the situation described in the problem. Label the relevant information: the tower, the wire, the angle of 52 degrees, the distance of 32 ft, and the angle of 21 degrees.
Step 2: Identify the right triangle
From the diagram, we can see that we have a right triangle formed by the wire, the ground, and the line connecting the point 32 ft from the base of the tower to the top of the tower.
Step 3: Define the variables
Let's define the following variables:
- h: height of the tower (what we want to find)
- x: distance from the base of the tower to the point on the ground
- L: total distance from the base of the tower to the point on the ground plus the 32 ft (x + 32)
Step 4: Apply trigonometry
Using trigonometry, we can write two equations involving the angles and the sides of the right triangle:
Equation 1: tan(52 degrees) = h / x
Equation 2: tan(21 degrees) = h / (x + 32)
Step 5: Solve the equations
We have two equations with two variables. We can solve them simultaneously to find the values of h and x.
From Equation 1, we can isolate h:
h = x * tan(52 degrees)
Substitute this expression for h into Equation 2:
tan(21 degrees) = (x * tan(52 degrees)) / (x + 32)
Now solve for x.
Multiply both sides by (x + 32):
(x + 32) * tan(21 degrees) = x * tan(52 degrees)
Expand and rearrange the equation to isolate x:
x * tan(21 degrees) + 32 * tan(21 degrees) = x * tan(52 degrees)
x * (tan(21 degrees) - tan(52 degrees)) = -32 * tan(21 degrees)
Finally, solve for x:
x = -32 * tan(21 degrees) / (tan(21 degrees) - tan(52 degrees))
Now that we know the value of x, we can substitute it back into Equation 1 to find h:
h = x * tan(52 degrees)
Step 6: Calculate the length of the tower
Finally, calculate the length of the tower by adding the height to the distance from the ground:
Length of tower = h + x
Substitute the values of h and x calculated in the previous steps to find the final answer.