PQ is a tower standing vertically on the horizontal ground. from A, the angle of elevation of top P of the tower is 45°. On moving 50 m. up a slope of 15°, the angle of elevation of P is found to be 75° from B. The horizobtal through B is BC. find height of the tower.
Well, well, well! It seems like we have a little geometric puzzle on our hands, don't we?
Let's break it down, my friend! We have a tower PQ standing vertically on the ground, and we have two different angles of elevation from two different points, A and B.
From point A, we have an angle of elevation of 45 degrees towards the top of the tower, P. And from point B, after moving 50 m up a slope of 15 degrees, we have an angle of elevation of 75 degrees towards point P.
Now, let's put on our geometry hats and figure this out. Are you ready?
First off, let's find the height of BC, which is the horizontal line passing through point B. We can use some trigonometry here. The height, BC, can be calculated by using the formula:
BC = 50 * tan(15)
Now that we have the height of BC, we can find the height of the tower! Let's call it h. It's time to do some more trigonometry.
We know that tan(75) = (h + BC) / 50
Rearranging this equation, we get:
h + BC = 50 * tan(75)
Substituting the value of BC we calculated earlier, we get:
h + 50 * tan(15) = 50 * tan(75)
Now, let's solve this equation and find the value of h. I bet your math skills are as sharp as a clown's nose, my friend!
Remember, while mathematics may not always be funny, we can always find humor in the way we approach it. So, put on a smile, crunch those numbers, and you'll find the height of the tower in no time!
To find the height of the tower, we can use trigonometry and set up two right triangles.
Let's label the points:
A - Point on the ground where the angle of elevation of the top of the tower is observed.
B - Point where the angle of elevation of the top of the tower is observed again after moving up the slope.
Given information:
Angle of elevation from A to the top of the tower (P) = 45°.
Angle of elevation from B to the top of the tower (P) = 75°.
Distance between A and B = 50 m.
Angle of slope = 15°.
Step 1: Calculate the height of point B above the ground.
In triangle BCP, B is the right angle, and angle C is equal to 15°. Since we know the distance BC, we can calculate the height of point B.
Using the trigonometric function tangent:
tan(15°) = height of B / BC
Rearranging the equation:
height of B = BC * tan(15°)
Calculating the height of B:
height of B = 50 * tan(15°) ≈ 12.99 m
Step 2: Calculate the height of the tower (P) above point B.
In triangle BXP, B is the right angle, and angle X = 75°. Since we know the height of B and the angle of elevation from B to P, we can calculate the height of P.
Using the trigonometric function tangent:
tan(75°) = height of P / BX
Rearranging the equation:
height of P = BX * tan(75°)
Calculating the height of P:
height of P = 12.99 * tan(75°) ≈ 62.5 m
Step 3: Calculate the total height of the tower (P).
The total height of the tower is the sum of the heights of B and P.
Total height of the tower = height of B + height of P = 12.99 m + 62.5 m ≈ 75.5 m
Therefore, the height of the tower is approximately 75.5 meters.
To find the height of the tower, we can use trigonometry. Let's break down the problem step by step:
Step 1: Drawing a diagram
Draw a diagram representing the given situation. Make sure to label all the known angles and distances accurately. It should include the tower, the angles of elevation, points A, B, C, and the distances mentioned.
Step 2: Determine key measurements
From the information given, we can identify the following measurements:
- Angle of elevation from A to the top of the tower (P) is 45°.
- Angle of elevation from B to the top of the tower (P) is 75°.
- Distance moved up the slope from B is 50 m.
- Angle of the slope is 15°.
Step 3: Setting up the trigonometric equations
Let's consider right-angled triangles PAC and PBC, where PA and PB are the heights of the tower.
In triangle PAC, we have:
- Angle PAC = 45° (given)
- Angle ACP = 90° (because PAC is a right-angled triangle)
In triangle PBC, we have:
- Angle PBC = 75° (given)
- Angle BPC = 90° (because PBC is a right-angled triangle)
Step 4: Applying trigonometric ratios
Using the trigonometric ratios, we can write the following equations:
For triangle PAC:
- tan(PAC) = PA / AC [using the tangent ratio]
For triangle PBC:
- tan(PBC) = PB / BC [using the tangent ratio]
Remember, tan(theta) = opposite/adjacent, where theta is an angle.
Step 5: Solving the equations
Using the given measurements, substitute the known values into the equations:
For triangle PAC: tan(45°) = PA / AC
- The tangent of 45° is 1, so it becomes: 1 = PA / AC
For triangle PBC: tan(75°) = PB / BC
- Calculate the tangent of 75° using a calculator, and substitute the value into the equation.
Step 6: Finding the heights
Simplify the equations to solve for PA (height of the tower) and PB (height from B):
From triangle PAC: 1 = PA / AC
- Rearrange the equation: PA = AC
From triangle PBC: tan(75°) = PB / BC
- Solve for PB: PB = BC * tan(75°)
Step 7: Calculate the height of the tower
Now, to find the height of the tower (PA), substitute the values from the triangle PBC into the equation: PA = AC.
Step 8: Calculate the height difference
The height difference between points A and B can be found by subtracting PA (height of the tower) from PB (height from B).
Step 9: Finalize the answer
The height of the tower is the height difference between A and B plus the distance moved up the slope from B.
By following these steps and executing the calculations, you should be able to find the height of the tower accurately.
Tan45 = h/d, h = d*Tan45.
Tan75 = h/(d-50), h = (d-50)*Tan75.
Therefore, h = d*Tan45 = (d-50)*Tan75.
d*Tan45 = (d-50)*Tan75.
divide both sides by Tan75:
0.268d = d-50,
d-0.268d = 50, d = 68.3 m.
h = d*Tan45 =
d = The hor. distance between point A and the foot of the tower.