A boy observes that the angle of elevation of the top of a tower is 32 degrees.He then walks 8m towards the tower and then discovers that the angle of elevation is 43 degrees.Find the height of the tower.
Sekou
d = h cot(32º)
d - 8 = h cot(43º)
8 = h [cot(32º) - cot(43º)]
h = 8 / [cot(32º) - cot(43º)]
To find the height of the tower, we can use trigonometry. Let's assume the height of the tower is "h" meters.
Step 1: Draw a diagram to visualize the situation.
T
|\
| \
h | \ x
| \
| \
|_____\
A B
In the diagram, T represents the top of the tower, A represents the initial position of the boy, and B represents the new position of the boy after walking 8m towards the tower.
Step 2: Identify the right-angled triangle in the diagram.
In this case, triangle TAB is the right-angled triangle.
Step 3: Use the tangent function to relate the angle of elevation, height of the tower, and the distance from the tower.
In triangle TAB, we have:
tan(32 degrees) = h / x (Equation 1)
tan(43 degrees) = h / (x - 8) (Equation 2)
Step 4: Simplify and solve the equations.
From Equation 1, we have:
h = x * tan(32 degrees)
From Equation 2, we have:
h = (x - 8) * tan(43 degrees)
Since both equations equal the same height "h", we can set them equal to each other and solve for "x".
x * tan(32 degrees) = (x - 8) * tan(43 degrees)
Step 5: Solve the equation for "x".
Using simple algebra, we can solve the equation:
x * tan(32 degrees) = x * tan(43 degrees) - 8 * tan(43 degrees)
x * (tan(32 degrees) - tan(43 degrees)) = -8 * tan(43 degrees)
x = (-8 * tan(43 degrees)) / (tan(32 degrees) - tan(43 degrees))
Step 6: Calculate the height of the tower.
Now that we have the value of "x", we can substitute it into Equation 1 to find the height "h" of the tower.
h = x * tan(32 degrees)
Substituting the value of "x" we found in Step 5, we can calculate:
h = [(-8 * tan(43 degrees)) / (tan(32 degrees) - tan(43 degrees))] * tan(32 degrees)
Now, evaluate this expression using a calculator to get the height of the tower.
To find the height of the tower, we can use trigonometry. Let's denote the height of the tower as "h" and the distance between the boy and the base of the tower as "x".
Step 1: Draw a diagram
Start by drawing a diagram to visualize the situation. Draw a vertical line to represent the tower and label it with "h". Place a right triangle where the top angle represents the angle of elevation (32 degrees initially and 43 degrees after the boy walks 8m). Label the horizontal distance from the initial position to the tower as "x" and the vertical distance from the initial position to the top of the tower as "h".
Step 2: Identify the trigonometric relationships
In the right triangle, the tangent of an angle is equal to the ratio of the opposite side to the adjacent side. Recall that the opposite side is the height of the tower and the adjacent side is the horizontal distance from the boy's initial position to the tower.
For the first observation, the tangent of the angle is given by tan(32) = h / x.
For the second observation, the tangent of the angle is given by tan(43) = h / (x - 8).
Step 3: Solve the equations
Now we have two equations:
1) tan(32) = h / x
2) tan(43) = h / (x - 8)
To solve these equations, we can use the method of substitution. Rearrange equation 1 to solve for h:
h = x * tan(32)
Substitute this value of h into equation 2:
tan(43) = (x * tan(32)) / (x - 8)
Step 4: Solve for x
Now we can solve the equation for x. Multiply both sides by (x - 8):
(x - 8) * tan(43) = x * tan(32)
Expand and rearrange:
x * tan(43) - 8 * tan(43) = x * tan(32)
Move all the terms involving x to one side:
x * tan(43) - x * tan(32) = 8 * tan(43)
Factor out the x:
x * (tan(43) - tan(32)) = 8 * tan(43)
Divide both sides by (tan(43) - tan(32)):
x = (8 * tan(43)) / (tan(43) - tan(32))
Step 5: Calculate the height (h)
Now that we have the value of x, we can substitute it back into equation 1 to find h:
h = x * tan(32)
Using the value of x from step 4, calculate:
h = (x from step 4) * tan(32)
By substituting the value of x from step 4 and evaluating the expression, you can find the value of h, which represents the height of the tower.