Three pegs R, S and T are on the vertices of a triangular plain field. R is 300 m from S on a bearing of 300° and T is 450 m directly south of R.
(i) the distance between T and S in meters:
(ii) the bearing of T from S.
(c) Find the area of the field, in hectares, correct to one decimal place.
Can you help me with this question Caleb is going to watch his brother play soccer on Saturday. Caleb asked his mother how long they will be away from home. Caleb's mother said they will drive 1/4 hour to get there, and they will drive another 1/4 hour back. Once they arrive, Caleb's brother will be playing in 3 soccer games. Each soccer game takes 3/4
hour, and there is no break in between games. Calculate how long will they be away from home.
To find the answers to these questions, we can break down the problem into smaller steps.
(i) Distance between T and S:
1. Draw a diagram of the triangular field.
2. Label the vertices R, S, and T.
3. Use the information given in the problem to determine the lengths of the sides of the triangle.
- R is 300 m from S on a bearing of 300°: This means that the distance between R and S is 300 m.
- T is 450 m directly south of R: This means that the distance between R and T is 450 m going directly south.
4. Use the Pythagorean theorem to find the distance between T and S.
- The distance between T and S is the hypotenuse of a right triangle with sides of lengths 450 m and 300 m.
- Apply the Pythagorean theorem: (distance between T and S)^2 = (distance between R and S)^2 + (distance between R and T)^2
- Substitute the given values: (distance between T and S)^2 = (300 m)^2 + (450 m)^2
- Solve for the distance between T and S by taking the square root of both sides.
5. Calculate the square root: distance between T and S = √[(300 m)^2 + (450 m)^2]
6. Simplify and calculate the value to find the distance between T and S in meters.
(ii) Bearing of T from S:
1. Use the diagram from step 3 of the previous part.
2. Find the bearing of T from S using the information given in the problem.
- In the problem, it is given that T is directly south of R.
- Since R is connected to S, directly south of R also means directly south of S.
- The bearing of a location is the angle measured clockwise from the north direction.
- In this case, since T is south of S, the bearing of T from S is 180°.
3. Therefore, the bearing of T from S is 180°.
(c) Area of the field:
1. Use the diagram from the previous parts.
2. To find the area of the triangular field, we need the base and height of the triangle.
- The base of the triangle is the distance between R and S, which is given as 300 m.
- The height of the triangle can be found by drawing a perpendicular line from T to the line segment RS.
3. Use the Pythagorean theorem to find the height of the triangle.
- The height of the triangle is the perpendicular distance from T to the line segment RS.
- Apply the Pythagorean theorem: (height)^2 + (distance between T and S)^2 = (distance between R and T)^2
- Substitute the given values: (height)^2 + (distance between T and S)^2 = (450 m)^2
- Solve for the height of the triangle by subtracting the square of the distance between T and S from both sides.
4. Calculate the square root: height = √[(450 m)^2 - (distance between T and S)^2]
5. Simplify and calculate the value to find the height of the triangle in meters.
6. Calculate the area of the triangle using the formula: Area = 0.5 * base * height.
7. Convert the area from square meters to hectares by dividing by 10,000 (as 1 hectare = 10,000 square meters).
8. Round the result to one decimal place.