Pregunta
- 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 .
of
(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
Responder
(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:
- Distance:
-
From Village Q to Town R:
- Distance:
km - Bearing:
- Distance:
-
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.
**(b)(i) The value of
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 .
**(b)(ii) The bearing of R from P is approximately
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Extra Insights
To illustrate Emefa’s journey, imagine a coordinate plane where you can plot points denoting her path:
- Start at point P (Village P).
- From P, draw a line at a 65° angle from North, and mark a point for Q (Village Q) at a distance of
km. - 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:
- From P to Q:
- 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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