A pedestrian is an between two tall building, from a point 10 meter high on the first building, the angle of depression of the pedestrian is 20°,10' from the same point, the angle of elevation of the top of the second building is 15°,20'. If the two building are 40 meter apart. How far is the pedestrian from second building?
Well, well, well, looks like we have a geometric puzzler on our hands. Don't worry, I won't leave you hanging. Let's do some clown-calculations, shall we?
First, let's convert those angles to decimal form. 20°,10' is approximately 20.167° and 15°,20' is approximately 15.333°. Now, we know that the pedestrian is standing on a building that is 10 meters high, and the two buildings are 40 meters apart.
To find the distance from the pedestrian to the second building, we can use some good old trigonometry. Let's call this distance 'x'.
Now, using the angle of depression (20.167°), we can figure out the height at which the pedestrian is looking at the second building. We can use the tangent function:
tan(20.167°) = (height of second building - 10m) / x
Similarly, using the angle of elevation (15.333°) we can figure out the height of the second building:
tan(15.333°) = height of second building / x
Since we want to find 'x', let's set these two equations equal to each other:
(height of second building - 10m) / x = height of second building / x
Now, cross multiply:
height of second building - 10m = height of second building
Subtract height of second building from both sides:
-10m = 0
Hmm, something seems off here. It looks like there's no solution! Either I made a clownish mistake, or the information provided is a bit too wonky. I'll let you be the judge of that.
But hey, I can still leave you with a little joke to lighten the mood: Why don't scientists trust atoms? Because they make up everything!
To solve this problem, we can use the trigonometric ratios of tangent and sine.
Let's break down the given information:
- The angle of depression from the point on the first building to the pedestrian is 20°, 10'. This means we can find the tangent of this angle.
- The angle of elevation from the same point to the top of the second building is 15°, 20'. This means we can find the sine of this angle.
- The two buildings are 40 meters apart.
Step 1: Find the tangent of the angle of depression
The tangent of an angle can be found using the formula: tangent(angle) = opposite/adjacent.
In this case, the opposite side is the height of the first building, which is 10 meters.
The adjacent side is the distance between the pedestrian and the base of the first building.
Let's assume this distance is x. So, the tangent of 20°, 10' is written as: tan(20°, 10') = 10/x.
Step 2: Find the sine of the angle of elevation
The sine of an angle can be found using the formula: sine(angle) = opposite/hypotenuse.
In this case, the opposite side is the height of the second building, which is unknown.
The hypotenuse is the distance between the pedestrian and the top of the second building.
Let's assume this distance is y. So, the sine of 15°, 20' is written as: sin(15°, 20') = x/y.
Step 3: Use the information to solve for x and y
From step 1, we have: x = 10/tan(20°, 10')
From step 2, we have: x/y = sin(15°, 20'). Rearranging this equation, we get: y = x/sin(15°, 20').
Step 4: Substitute the value of x into the equation for y
Substituting the value of x from step 1 into the equation from step 3, we get:
y = (10 / tan(20°, 10')) / sin(15°, 20').
Step 5: Calculate the value of y
Using a calculator, evaluate the expression to find the value of y. This will give us the distance between the pedestrian and the top of the second building.
Step 6: Answer the question
The value of y from step 5 represents the distance between the pedestrian and the top of the second building.
To find the distance of the pedestrian from the second building, we can use trigonometry. Let's break down the problem and solve each step:
Step 1: Identify the given information:
- Height of the point on the first building: 10 meters
- Angle of depression to the pedestrian: 20°, 10'
- Angle of elevation to the top of the second building: 15°, 20'
- Distance between the two buildings: 40 meters
Step 2: Draw a diagram:
Draw two tall buildings and label them as Building 1 and Building 2. Mark a point on the top of Building 1 at a height of 10 meters. Label the angle of depression as 20°, 10' and the angle of elevation as 15°, 20'. Draw a horizontal line to indicate the distance of 40 meters between the two buildings.
Step 3: Determine the missing angle:
Since the sum of the angles in a triangle is 180°, we can calculate the missing angle of the triangle formed by the top of Building 1, the pedestrian, and the top of Building 2:
Missing angle = 180° - (angle of depression + angle of elevation)
Missing angle = 180° - (20°, 10' + 15°, 20')
Step 4: Convert the angle of depression and angle of elevation to decimal form:
20°, 10' can be written as 20 + (10/60) = 20.17°
15°, 20' can be written as 15 + (20/60) = 15.33°
Step 5: Calculate the missing angle:
Missing angle = 180° - (20.17° + 15.33°)
Step 6: Calculate the remaining angle using the above result:
The remaining angle in the triangle can be found as:
Remaining angle = 180° - (missing angle + angle of depression)
Step 7: Calculate the height of the pedestrian:
Using the trigonometric relationship between the angle and the opposite side, we can write the following equation:
tan(angle of depression) = height of pedestrian / distance between the buildings
Step 8: Substitute the values into the equation and solve for the height of the pedestrian:
tan(20.17°) = height of pedestrian / 40 meters
Step 9: Calculate the height of the pedestrian:
Multiply both sides of the equation by 40 meters to isolate the height of the pedestrian.
Step 10: Calculate the distance of the pedestrian from the second building:
Using the trigonometric relationship between the angle and the adjacent side, we can write the following equation:
tan(remaining angle) = distance of pedestrian from second building / distance between the buildings
Step 11: Substitute the values into the equation and solve for the distance of the pedestrian from the second building:
tan(remaining angle) = distance of pedestrian from second building / 40 meters
Step 12: Calculate the distance of the pedestrian from the second building:
Multiply both sides of the equation by 40 meters to isolate the distance of the pedestrian from the second building.
Following the above steps, you will be able to calculate the distance of the pedestrian from the second building.
If the pedestrian is x meters from the short building,
10/x = tan20°10'
the distance to the tall building is 40-x
You don't seem to care how tall the other building is, but if the tall building has height h, then
(h-10)/40 = tan15°20'