a plat form and a building are on the same horizontal plane , the angle of depression of the bottom of the building from the top A of the platform is 39 degree. the angle of elevation of the top D of the building from the top of the platform is 56 degree. given that the distance between the foot of the platform and that of the building is 10m. calculate the height of the building, to the nearest whole number
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What is the answer?
To solve this problem, let's define the following points:
A - Top of the platform
B - Bottom of the building
C - Foot of the platform
D - Top of the building
From the given information, we know that angle BAD is 39 degrees (angle of depression) and angle DAB is 56 degrees (angle of elevation). We also know that the distance between point C and point B is 10 meters.
Step 1: Find the height of the platform (CB).
Using trigonometry, we can use the tangent function:
tan(DAB) = CB / AC
tan(56) = CB / AC
CB = AC * tan(56)
Step 2: Find the distance between the top of the platform and the bottom of the building (AD).
Using trigonometry, we can use the tangent function:
tan(BAD) = AD / AC
tan(39) = AD / AC
AD = AC * tan(39)
Step 3: Find the height of the building (DB).
DB = CB + AD
Step 4: Substitute the values and calculate.
AC = 10 meters (given)
CB = AC * tan(56)
= 10 * tan(56)
≈ 10 * 1.4389
≈ 14.3889 meters
AD = AC * tan(39)
= 10 * tan(39)
≈ 10 * 0.8090
≈ 8.0900 meters
DB = CB + AD
= 14.3889 + 8.0900
≈ 22.4789 meters
Therefore, the height of the building is approximately 22 meters to the nearest whole number.
To find the height of the building, we can use trigonometric ratios and create a triangle with the given information.
Let's label the points as follows:
- The bottom of the building: B
- The top of the platform: A
- The top of the building: D
- The foot of the platform: C
We know that angle BAC (angle of depression) is 39 degrees, and angle CAD (angle of elevation) is 56 degrees. The distance between the foot of the platform (point C) and the building (point B) is given as 10m.
Now, let's find the distance from the top of the platform (point A) to the top of the building (point D). We'll call this distance x.
In triangle BAC, we can use the tangent function:
tan(angle BAC) = opposite/adjacent
tan(39°) = BC/AC
We know AC = 10m, so we can solve for BC:
BC = tan(39°) * 10
Next, in triangle CAD, we can use the tangent function:
tan(angle CAD) = opposite/adjacent
tan(56°) = BD/(AC + x)
We know BD = BC + CD, and AC = 10m, so we can rewrite the equation:
tan(56°) = (BC + CD)/(10 + x)
Now we can substitute the value of BC from the previous step:
tan(56°) = (tan(39°) * 10 + CD)/(10 + x)
Simplifying this equation, we have a single equation with one unknown (x) and CD:
tan(56°) = (tan(39°) * 10 + CD)/(10 + x)
We can now solve this equation for CD. Rearranging the equation, we get:
CD = (10 + x) * tan(56°) - tan(39°) * 10
Finally, we can use CD to find the height of the building (DB):
DB = CD + BC
Substituting the value of CD, we have:
DB = (10 + x) * tan(56°) - tan(39°) * 10 + tan(39°) * 10
Now, you can substitute the values of tan(56°) and tan(39°) into the equation to calculate the height of the building (DB). Rounding the result to the nearest whole number will give you the final answer.