Pregunta
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  1. Emefa walked from a village, P to a friend in another village, Q on a bearing
    of . After spending some time with her friend, she continued to a nearby town, R ,
    away on a bearing of . If the distance between P and R is :
    (a) illustrate the information in a diagram;
    (b) calculate, correct to the nearest whole number, the:
    (i) value of ;
    (ii) bearing of R from P .

Ask by Lang Griffin. in Ghana
Jan 05,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

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(b)(i) The value of is 6 km.
(b)(ii) The bearing of R from P is approximately .

Solución

Let’s solve the problem step by step.

Given:

  • From Village P to Village Q:
    • Distance: km
    • Bearing:
  • From Village Q to Town R:
    • Distance: km
    • Bearing:
  • Distance between Village P and Town R (PR): km

Part (b)(i): Calculating the Value of

**1. Establish a Coordinate System:
  • Let’s place Village P at the origin .
  • Bearings are measured clockwise from the north (positive y-axis).
**2. Determine Coordinates of Village Q:
  • Bearing of :
  • Approximations:
  • Coordinates:
**3. Determine Coordinates of Town R:
  • Bearing of :
  • Approximations:
  • Coordinates:
**4. Calculate Distance PR:
  • Squaring both sides:
Answer:
**(b)(i) The value of is 6 km.

Part (b)(ii): Calculating the Bearing of R from P

**1. Determine Coordinates with :
  • km
  • km
**2. Calculate the Bearing:
  • The bearing is the angle measured clockwise from the north.
**3. Compute the Angle:
**4. Determine the Bearing:
  • Since is negative and is positive, Town R is in the southeast quadrant.
  • Bearing
Answer:
**(b)(ii) The bearing of R from P is approximately .

Respondido por UpStudy AI y revisado por un tutor profesional

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Extra Insights

To illustrate Emefa’s journey, imagine a coordinate plane where you can plot points denoting her path:
  1. Start at point P (Village P).
  2. From P, draw a line at a 65° angle from North, and mark a point for Q (Village Q) at a distance of km.
  3. From Q, draw a line at a 155° angle from North towards point R (Town R), marking this at a distance of km.
Now, to solve for :
Using the Law of Cosines on triangle PQR where PR = :
Calculate the angle QPR using bearings:
. Hence, cos(90°) = 0.
So rearranging gives:
Now, for part (ii), calculating the bearing of R from P involves finding the angles to add:
  1. From P to Q:
  2. From Q to R: . Thus, the concept of bearings implies an adjustment considering clockwise from North.
Now, summing these gives:
So, the bearing of R from P is .

Fun Fact: The Law of Cosines, or if you like, the “Triangle’s Secret Sauce”, is used for solving those tricky triangle problems where you don’t have a right angle to work with, often stealing the spotlight in geometry class!
Further Reading: If you’re intrigued by bearings and triangle properties, try delving into trigonometry textbooks or online courses that cover navigation and surveying techniques—they’re like treasure maps for the mathematically adventurous!

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