A water tower is located 354 feet from a building. From a window in the building, an observer notes that the angle of elevation to the top of the tower is 42° and that the angle of depression to the bottom of the tower is 20°.
How tall is the tower?
How high is the window from the ground?
An isosceles triangle has an area of 24 yd², and the angle between the two equal sides is 166°. Find the length of the two equal sides.
It's easier to solve if you draw the figure. If you're not sure what 'angle of elevation' or 'angle of depression' is about, I suggest you read some lectures about it first.
In your drawing, you can separate them into two right triangles.
One triangle has an angle of 20°, with a height of 354 feet. We can solve for its base using tangents:
tan(angle) = opposite / adjacent
tan(20°) = a / 354
354*tan(20°) = a
a = 128.85 ft
The other triangle has an angle of 42°, with height of 354 feet. We can also solve for its base using tangents:
tan(angle) = opposite / adjacent
tan(42°) = b / 354
354*tan(42°) = b
b = 318.74 ft
The height of window is equal to a.
To get the height of tower, get the sum of a and b: a + b = ?
Hope this helps~ `u`
To solve this problem, we can use trigonometry and the properties of right-angled triangles. Here's how you can find the height of the tower and the height of the window from the ground:
1. Let's start by considering the angle of elevation. The observer sees the top of the tower from the window. We have a right-angled triangle with the observer at the window, the top of the tower, and a point on the ground directly below the observer (call it point A).
2. We can label the height of the tower as h and the distance from the building to the tower as x. Using the angle of elevation, we can set up the equation:
tan(42°) = h / x
3. Next, let's consider the angle of depression. The observer sees the bottom of the tower from the window. We have another right-angled triangle with the same observer, the bottom of the tower, and point A.
4. From this angle, we know that the angle of depression is 20°. We can use this information to set up another equation:
tan(20°) = h / (x + 354)
5. Now, we have a system of two equations with two variables (h and x). We can solve this system of equations to find the height of the tower (h) and the distance from the building to the tower (x).
6. Rearranging the first equation, we get:
x = h / tan(42°)
7. Substituting this value of x into the second equation, we have:
tan(20°) = h / (h / tan(42°) + 354)
8. Simplifying the equation above, we can solve for h:
h ≈ tan(20°) * (h / tan(42°) + 354)
9. To solve for h, let's isolate h on one side of the equation. Multiply both sides of the equation by (h / tan(42°) + 354):
h ≈ tan(20°) * h + tan(20°) * 354
10. Next, move the term tan(20°) * h to one side of the equation:
h - tan(20°) * h ≈ tan(20°) * 354
11. Factor out h on the left side of the equation:
h * (1 - tan(20°)) ≈ tan(20°) * 354
12. Divide both sides of the equation by (1 - tan(20°)):
h ≈ (tan(20°) * 354) / (1 - tan(20°))
13. Now, you can plug the value of tan(20°) into your calculator and calculate the value of h.
h ≈ (tan(20°) * 354) / (1 - tan(20°))
14. Once you have calculated the height of the tower (h), you can find the height of the window from the ground by subtracting it from h.
Please plug in the values into the equation, and you will get the actual height of the tower and the height of the window from the ground.