A tower stands on a 20 degree slope. From a point 45 m directly up the slope. The angle of elevation of the top of the tower is 10 degrees. How tall is the tower?
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To find the height of the tower, we can use trigonometry. Let's break down the problem step by step:
Step 1: Draw a diagram
It's always helpful to visualize the problem. Draw a diagram with a tower, a slope, and a point 45m up the slope.
Step 2: Identify the known values
From the problem, we know:
- The angle of elevation of the top of the tower is 10 degrees.
- The distance directly up the slope is 45m.
- The slope angle is 20 degrees.
Step 3: Determine the component of the distance directly up the slope
Since the angle of elevation is 10 degrees, it means the vertical height is a component of the total distance directly up the slope. To find this component, we can use trigonometry.
The component of the distance directly up the slope can be calculated using the formula:
Component = Total Distance * sin(Angle)
In this case, the component is:
Component = 45m * sin(10 degrees)
Step 4: Calculate the height of the tower
The height of the tower is equal to the component of the distance directly up the slope plus the vertical height of the slope.
The vertical height of the slope can be calculated using the formula:
Vertical Height = Distance * sin(Angle)
In this case, the vertical height of the slope is:
Vertical Height = 45m * sin(20 degrees)
Finally, to find the height of the tower, we add the component and the vertical height together:
Tower Height = Component + Vertical Height
Step 5: Substitute the values and calculate
Let's substitute the values:
Component = 45m * sin(10 degrees)
Vertical Height = 45m * sin(20 degrees)
Now, calculate the Component and Vertical Height values, and then add them to find the Tower Height.
Once you have calculated the height using the trigonometric calculations, you will have your answer for how tall the tower is.