solve---**Question Identifier:**
Question 2
**Textual Information:**
Talia finds that the angle of elevation of the top of a pole, A from B is 30°, but when she walks 30 metres towards it to point P, the angle of elevation is 45°.
(a) Show that AP = (30 sin 30°) / sin 15°.
(b) Hence, calculate the height of the pole correct to the nearest metre.
**Chart/Diagram Description:**
* **Type:** Geometric figure, specifically a combination of two right-angled triangles sharing a common vertical side.
* **Main Elements:**
* **Points:** Four labeled points: A, B, C, P.
* **Lines/Segments:**
* A vertical line segment AC, representing the pole, with its length labeled as 'x'.
* A horizontal line segment BC, representing the ground.
* A point P lies on BC, between B and C.
* A line segment AB connecting point A to point B.
* A line segment AP connecting point A to point P.
* **Angles:**
* Angle ABC (angle of elevation from B to A) is labeled as 30°.
* Angle APC (angle of elevation from P to A) is labeled as 45°.
* Angle ACB (angle at the base of the pole) is marked with a right-angle symbol (90°). This implies triangle APC and triangle ABC are right-angled at C.
* **Lengths:**
* The distance between B and P (length of segment BP) is labeled as 30.
* The height of the pole (length of segment AC) is labeled as 'x'.
* **Relative Position:** Point A is vertically above point C. Points B, P, and C are collinear and lie on a horizontal line, with P between B and C.
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We have an angle of elevation problem. Talia observes a pole from point B with an angle of elevation of 30 degrees. She then walks 30 meters closer to point P, where the angle of elevation becomes 45 degrees. We need to prove that AP equals 30 sine 30 degrees divided by sine 15 degrees, and then calculate the height of the pole.
Let's analyze the two right triangles formed in this problem. Triangle ABC has a right angle at C, with angle ABC equal to 30 degrees. Using trigonometry, tangent of 30 degrees equals x over BC. Triangle APC also has a right angle at C, with angle APC equal to 45 degrees. Here, tangent of 45 degrees equals x over PC. The key relationship is that BC equals PC plus 30 meters, since Talia walked 30 meters from B to P.
Now let's derive the key equations. From triangle APC, since tangent of 45 degrees equals 1, we get PC equals x. From triangle ABC, tangent of 30 degrees equals 1 over root 3, so BC equals x times root 3. Since BC equals PC plus 30, we have x root 3 equals x plus 30. Solving for x, we get x equals 30 over root 3 minus 1. To rationalize this expression, we multiply by root 3 plus 1 over root 3 plus 1, giving us x equals 15 times root 3 plus 1.
Now we'll prove the formula for AP using the Law of Sines in triangle ABP. First, let's identify the angles. Angle ABP is 30 degrees, angle APB is 180 minus 45 which equals 135 degrees, and angle BAP is 180 minus 30 minus 135 which equals 15 degrees. Next, we apply the Law of Sines: BP over sine of angle BAP equals AP over sine of angle ABP. Substituting our values: 30 over sine of 15 degrees equals AP over sine of 30 degrees. Solving for AP, we get AP equals 30 sine 30 degrees over sine 15 degrees, which completes the proof of part a.
Now let's calculate the numerical height of the pole. First, we calculate AP using our proven formula. Since sine of 30 degrees equals 0.5 and sine of 15 degrees is approximately 0.2588, we get AP equals 30 times 0.5 divided by 0.2588, which gives us approximately 57.74 meters. Next, we use triangle APC where sine of 45 degrees equals x over AP. Since sine of 45 degrees equals root 2 over 2, approximately 0.7071, we calculate x equals 57.74 times 0.7071, which gives us approximately 40.8 meters. Therefore, the height of the pole is 41 meters to the nearest meter.