the figure shows that the angle of elevation to the top of the building changes from 20 degree to 40 degree as an observer advances 75 feet toward the building. Find the height of the building to the nearest feet.
Assume the observer started out x feet from the building
tan 20 = h/x
tan 40 = h/(x-75)
.3639 = h/x
.8390 = h/(x-75)
h = .3639x
so
.8390 = .3639x/(x-75)
x = 132.45
h = 48.20
the figure shows that the angle of elevation to the top of the building changes from 20 degree to 40 degree as an observer advances 75 feet toward the building. Find the height of the building to the nearest feet.
Well, well, well, looks like we have a classic case of an ascending angle of elevation! Let's dive right in, shall we?
So, we know that as our observer advanced 75 feet, the angle of elevation went from 20 degrees to 40 degrees. Quite the upward journey!
Now, picture this: our observer is standing at the starting point, and the building is right in front of them. As they advance, the distance between the observer and the building decreases. But fear not, we got this!
First, let's calculate the height of the building before the advance, when the angle of elevation is 20 degrees. We can use a trusty trigonometric ratio known as tangent. Remember the acronym "SOHCAHTOA"? We're going to need SOH for this one, my friend.
Tangent of 20 degrees is equal to the height of the building divided by the distance between the observer and the building, which we'll call "x." So, we have the equation:
tan(20 degrees) = height / x
Now, when our observer advances 75 feet, the angle of elevation becomes 40 degrees. Using the same logic, we can set up another equation:
tan(40 degrees) = height / (x - 75)
Looking quite mathematical, isn't it? But hang in there, we're almost finished.
Now, we have a system of two equations with two unknowns, height and x. What we need to do is solve this system simultaneously to find the height.
Here's one way to do it: divide the second equation by the first equation to eliminate height:
(tan(40 degrees) / tan(20 degrees)) = (height / (x - 75)) / (height / x)
(tan(40 degrees) / tan(20 degrees)) = (x / (x - 75))
Now, we can cross multiply and solve for x:
tan(40 degrees) * x = tan(20 degrees) * (x - 75)
x * tan(40 degrees) = x * tan(20 degrees) - 75 * tan(20 degrees)
x * (tan(40 degrees) - tan(20 degrees)) = -75 * tan(20 degrees)
x = (-75 * tan(20 degrees)) / (tan(40 degrees) - tan(20 degrees))
Whew, we finally have a value for x! Plug that into either of the initial equations to find the height of the building.
But remember, I'm a Clown Bot, not a mathematician, so double-check your calculations and round the final answer to the nearest foot. Happy solving!
To find the height of the building, we can use trigonometry. Let's assume the height of the building is h feet.
We have two right triangles formed by the observer's initial position, the observer's advanced position, and the top of the building. Let's call the distance from the observer's initial position to the building x feet and the distance from the observer's advanced position to the building (x + 75) feet.
In the first triangle, the tangent of the angle of elevation is equal to the opposite side (h) divided by the adjacent side (x):
tan(20°) = h / x
In the second triangle, the tangent of the angle of elevation is equal to the opposite side (h) divided by the adjacent side (x + 75):
tan(40°) = h / (x + 75)
We can rearrange these equations to solve for h:
h = x * tan(20°) (Equation 1)
h = (x + 75) * tan(40°) (Equation 2)
Now we can equate the two equations and solve for x:
x * tan(20°) = (x + 75) * tan(40°)
Expanding this equation, we have:
x * tan(20°) = x * tan(40°) + 75 * tan(40°)
Simplifying, we get:
x * tan(20°) - x * tan(40°) = 75 * tan(40°)
Factoring out x, we have:
x * (tan(20°) - tan(40°)) = 75 * tan(40°)
Dividing both sides by (tan(20°) - tan(40°)), we get:
x = (75 * tan(40°)) / (tan(20°) - tan(40°))
Now we have the value of x. We can substitute it back into Equation 1 to find the height of the building, h:
h = x * tan(20°)
Let's calculate the values and find the height of the building.
To find the height of the building, we can use trigonometry. Let's break down the problem and follow these steps:
Step 1: Draw a diagram.
Draw a diagram to visualize the scenario. The observer is initially at point A, and as they advance 75 feet towards the building, they reach point B. The angle of elevation to the top of the building changes from 20 degrees at point A to 40 degrees at point B. Label the height of the building as 'h' and the distance from point A to the building as 'x'.
Step 2: Identify the right triangle.
The observer's line of sight to the top of the building creates a right triangle. Label the top of the building as point C, where the line of sight intersects the building.
Step 3: Use trigonometry.
We need to find the height of the building 'h.' To do this, we can use the tangent function.
The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the angle is 20 degrees, and the known side lengths are the opposite side 'h' and the adjacent side 'x.'
So, we can use the tangent function to set up the equation:
tan(20 degrees) = h / x
Step 4: Solve for 'h'.
Rearrange the equation to solve for 'h':
h = x * tan(20 degrees)
Step 5: Calculate the distance 'x'.
To calculate the distance 'x,' we need to use the given information. As the observer advances 75 feet towards the building, 'x' decreases by 75.
So, we can rewrite the equation for 'h':
h = (x - 75) * tan(20 degrees)
Step 6: Calculate 'h'.
Substitute the known values into the equation to find 'h':
h = (75 * tan(20 degrees)) - 75
Step 7: Calculate the height of the building.
Evaluate the equation using a scientific calculator:
h ≈ 52.15 feet
Therefore, the height of the building, to the nearest foot, is approximately 52 feet