Matthew parks his car between Karen’s and Patrick’s apartment buildings. The car is 46 m in front of Karen’s apartment building. The angle of elevation from the car to the top of the building is 40°. Matthew’s car is 39 m away from Patrick’s apartment building. The angle of elevation from the car to the top of the building is 50°.
a) Determine the height of each building, to the nearest metre.
b) State which building is taller, and by how much.
if the buildings are of height k and p,
k/46 = tan40°
p/39 = tan50°
now just evaluate k and p, take the difference
Take the tangent of Karen’s and Matthew’s building to determine the height. Then subtract the two heights
To solve this problem, we can use trigonometric ratios in right triangles. Let's start solving step by step:
Step 1: Determine the height of Karen's apartment building.
We have a right triangle with the car, Karen's apartment building, and the line connecting the car to the top of the building. Let's call the height of Karen's apartment building "h1".
In this triangle, we have the opposite side (h1), the adjacent side (46 m), and the angle of elevation (40°). We can use the tangent function to find the height:
tan(40°) = h1 / 46
h1 = tan(40°) * 46
Using a calculator, we find h1 ≈ 43.34 m.
Therefore, the height of Karen's apartment building is approximately 43 meters.
Step 2: Determine the height of Patrick's apartment building.
Similar to step 1, let's call the height of Patrick's apartment building "h2".
In this triangle, we have the opposite side (h2), the adjacent side (39 m), and the angle of elevation (50°). Again, we can use the tangent function to find the height:
tan(50°) = h2 / 39
h2 = tan(50°) * 39
Using a calculator, we find h2 ≈ 49.96 m.
Therefore, the height of Patrick's apartment building is approximately 50 meters.
Step 3: Determine which building is taller and by how much.
Comparing the heights of Karen's and Patrick's apartment buildings, we find h2 > h1.
Therefore, Patrick's apartment building is taller than Karen's.
The difference in height is h2 - h1 = 50 m - 43 m = 7 meters.
Therefore, Patrick's apartment building is taller than Karen's by 7 meters.
To solve this problem, we can use the concept of trigonometry, specifically tangent function, to find the height of each building.
a) To find the height of Karen's apartment building, we can use the formula: height = distance * tangent(angle of elevation).
Let's calculate:
For Karen's building:
distance = 46 m
angle of elevation = 40°
Using the tangent function: tangent(40°) = height / 46
Rearranging the equation: height = 46 * tangent(40°)
Using a calculator, calculate the tangent of 40°: tan(40°) ≈ 0.8391
height ≈ 46 * 0.8391
height ≈ 38.5786
So, the height of Karen's apartment building is approximately 39 meters.
(rounded to the nearest meter)
For Patrick's building:
distance = 39 m
angle of elevation = 50°
Using the tangent function: tangent(50°) = height / 39
Rearranging the equation: height = 39 * tangent(50°)
Using a calculator, calculate the tangent of 50°: tan(50°) ≈ 1.1918
height ≈ 39 * 1.1918
height ≈ 46.5582
So, the height of Patrick's apartment building is approximately 47 meters.
(rounded to the nearest meter)
b) To determine which building is taller, we can compare the heights we found above. Since Patrick's building has a height of 47 meters and Karen's building has a height of 39 meters, we can conclude that Patrick's building is taller by 8 meters.
Therefore, Patrick's building is taller than Karen's building by approximately 8 meters.