A contractor needs to know the height of a building to estimate the cost of a job. From a point 84 feet away from the base of the building, the angle of elevation to the top of the building is found to be 44 degrees and 23'. Find the height of the building. Round your answer to the hundredths place.
h/84 = tan 44 deg 23 min.
h = 82.21 ft
Well, the contractor definitely needs to know the height of the building, or else they might just end up building a really tall mailbox instead. Let's get to solving this fun little problem, shall we?
First things first, we need to convert that 44 degrees and 23 minutes into decimal form. Converting minutes to degrees, we get:
23 minutes * (1 degree / 60 minutes) = 0.3833 degrees
Now let's add that to the initial 44 degrees:
44 degrees + 0.3833 degrees = 44.3833 degrees
Phew, now that we've got that out of the way, it's time to bring out our trusty trigonometry. We have a right triangle here, with the height of the building as the opposite side and the distance from the building as the adjacent side. The tangent of the angle of elevation is defined as:
tangent(angle) = opposite / adjacent
Since we're looking for the height (opposite side), we rearrange the formula:
opposite = tangent(angle) * adjacent
Now let's plug in the values we have:
tangent(44.3833 degrees) = height / 84 feet
To solve for the height, we multiply both sides by 84 feet:
height = tangent(44.3833 degrees) * 84 feet
Now, let's grab our calculators and do the math:
height = tan(44.3833 degrees) * 84
height ≈ 72.45 feet
So, the estimated height of the building is approximately 72.45 feet. We wouldn't want the contractor to end up with a skyscraper instead of a single-story house, now would we?
To find the height of the building, we can use trigonometry.
Step 1: Convert the angle of elevation from degrees and minutes to decimal degrees.
- Since there are 60 minutes in a degree, we need to divide 23 minutes by 60 to get the decimal part.
- 23 / 60 = 0.3833 (rounded to four decimal places)
- So the angle of elevation is 44.3833 degrees.
Step 2: Set up the trigonometric equation using the tangent function.
- Tangent is calculated as the opposite side divided by the adjacent side.
- In this case, the opposite side is the height of the building, and the adjacent side is the distance from the base of the building to the point of observation.
- tan(angle) = opposite / adjacent
- tan(44.3833°) = opposite / 84 feet
Step 3: Solve for the height of the building (opposite side).
- Multiply both sides of the equation by 84 feet to isolate the opposite side.
- opposite = tan(44.3833°) * 84
- opposite = 60.15 feet (rounded to two decimal places)
Therefore, the height of the building is approximately 60.15 feet.
To find the height of the building, we can use trigonometry. We have the angle of elevation and the distance from the base of the building to the point of observation.
First, let's convert the angle of elevation to decimal degrees. There are 60 minutes in 1 degree, so angle in degrees = (44 + (23/60)).
Therefore, the angle in degrees = 44.3833 degrees (rounded to four decimal places).
Now, we can use the tangent function to find the height of the building. The tangent of an angle is equal to the ratio of the opposite side (height of the building) to the adjacent side (distance from the base of the building).
In this case, we need to find the height, so let's denote it as 'h'.
We also have the adjacent side as 84 feet.
Using the tangent function, we can write the equation as follows: tan(angle) = opposite/adjacent.
Therefore, tan(44.3833) = h/84.
Now, we can solve for 'h' by rearranging the equation:
h = tan(44.3833) * 84.
Using a calculator, we can multiply tan(44.3833) by 84:
h ≈ 73.17 feet.
Therefore, the height of the building is approximately 73.17 feet.