A cross- country skier skis 7.40km in the direction of 45⁰ East of South, then 2.8 km in the direction 30⁰ North of East, and finally 5.2km in the direction 22⁰ West of North. How far is the shier from the starting point?
d = 40km[315o]+2.8km[30o]+5.2km[112o]
X = 28.28 + 2.425 - 1.948 = 28.76 km.
Y = -28.28 + 1.4 + 4.821 = -22.06 km.
d = sqrt(28.76^2+(-22.06)^2) = 36.2 km.
NOTE: To find X, multiply by the cosine;
to find Y, multiply by the sine.
To determine the skier's final position, we can break down the distances and directions into their respective north and east components.
1. The skier travels 7.40 km in the direction of 45⁰ East of South. This can be split into its north and east components using trigonometry:
- East component: 7.40 km * cos(45⁰) = 7.40 km / √2.
- South component: 7.40 km * sin(45⁰) = 7.40 km / √2.
2. Next, the skier travels 2.8 km in the direction of 30⁰ North of East. This can be split into its north and east components as follows:
- East component: 2.8 km * cos(30⁰) = 2.8 km * (√3 / 2) = 2.8 km * √3 / 2.
- North component: 2.8 km * sin(30⁰) = 2.8 km * (1 / 2) = 1.4 km.
3. Finally, the skier travels 5.2 km in the direction 22⁰ West of North. This can be split into its north and east components:
- West component: 5.2 km * cos(22⁰) = 5.2 km * (√(1 - (sin(22⁰))^2)).
- North component: 5.2 km * sin(22⁰) = 5.2 km * sin(22⁰).
To find the skier's final position, we can add up the north and east components:
- East component: 7.40 km / √2 + 2.8 km * √3 / 2.
- North component: 7.40 km / √2 + 1.4 km + 5.2 km * sin(22⁰).
Using the calculated values, plug them into the equation to find the final position:
- East component: 7.40 km / √2 + 2.8 km * √3 / 2.
- North component: 7.40 km / √2 + 1.4 km + 5.2 km * sin(22⁰).
Compute the east component and north component to find the final position of the skier.
To find the total distance from the starting point, we need to add up the distances traveled in each direction.
First, let's break down the distances into their components (north/south and east/west):
For the first leg:
- Distance: 7.40 km
- Direction: 45⁰ East of South
To find the distance traveled in the north/south direction, we can use trigonometry. The angle between the skier's direction and the north/south line is (90⁰ - 45⁰) = 45⁰. So the component in the north/south direction is given by:
Distance_N/S = 7.40 km * sin(45⁰)
To find the distance traveled in the east/west direction, we can also use trigonometry. The angle between the skier's direction and the east/west line is 45⁰. So the component in the east/west direction is given by:
Distance_E/W = 7.40 km * cos(45⁰)
Next, let's calculate the components for the second and third legs:
For the second leg:
- Distance: 2.8 km
- Direction: 30⁰ North of East
Distance_N/S = 2.8 km * cos(30⁰)
Distance_E/W = 2.8 km * sin(30⁰)
For the third leg:
- Distance: 5.2 km
- Direction: 22⁰ West of North
Distance_N/S = 5.2 km * cos(22⁰)
Distance_E/W = 5.2 km * sin(22⁰)
Now, let's add up all the north/south and east/west components separately:
Total_N/S = Distance_N/S (first leg) + Distance_N/S (second leg) + Distance_N/S (third leg)
Total_E/W = Distance_E/W (first leg) + Distance_E/W (second leg) + Distance_E/W (third leg)
Finally, we can find the total distance from the starting point using the Pythagorean theorem:
Total Distance = sqrt(Total_N/S^2 + Total_E/W^2)
Simply plug in the values for each component and calculate to get the final answer.