An orienteer runs for 4.5km, then turns through and angle of 32 degrees and runs for another 6km.How far is she from her starting point?
x^2 = 4.5^2 + 6^2 - 2(4.5)(6)cos148°
x = 10.1 km
Well, let me put on my running shoes and calculate this for you. The orienteer ran for 4.5km, turned 32 degrees, and then ran another 6km. I guess you could say she took a little detour. Anyway, let's find out how far she is from her starting point. According to my calculations, she's approximately "off track" by 0.636 km. So, if we round it to two decimal places, she is about 0.64 km from her starting point. Looks like she needs a compass for her next adventure!
To solve this problem, we can use trigonometry and the concept of a right triangle.
Step 1: Calculate the horizontal displacement (X)
The horizontal displacement can be calculated using the formula:
X = adjacent side of the right triangle = cos(angle) * hypotenuse
In this case, the adjacent side is the horizontal displacement, the angle is 32 degrees, and the hypotenuse is 6 km.
X = cos(32 degrees) * 6 km
Step 2: Calculate the vertical displacement (Y)
The vertical displacement can be calculated using the formula:
Y = opposite side of the right triangle = sin(angle) * hypotenuse
In this case, the opposite side is the vertical displacement, the angle is 32 degrees, and the hypotenuse is 6 km.
Y = sin(32 degrees) * 6 km
Step 3: Calculate the total displacement (D) using the Pythagorean theorem
The total displacement can be calculated using the formula:
D = square root of (X^2 + Y^2)
In this case, we have the horizontal displacement (X) and the vertical displacement (Y).
D = square root of (X^2 + Y^2)
Now, let's calculate the values:
Step 1:
X = cos(32 degrees) * 6 km
X ≈ 5.027 km
Step 2:
Y = sin(32 degrees) * 6 km
Y ≈ 3.099 km
Step 3:
D = square root of (X^2 + Y^2)
D = square root of (5.027^2 + 3.099^2)
D ≈ 5.981 km
Therefore, the orienteer is approximately 5.981 km away from her starting point.
To find the distance from the starting point, we can use the concept of vectors and trigonometry.
Step 1: Find the horizontal and vertical components of each leg of the orienteer's journey.
The horizontal component is given by the formula: distance x cos(angle)
The vertical component is given by the formula: distance x sin(angle)
For the first leg of the journey:
Horizontal component = 4.5 km x cos(0°) = 4.5 km
Vertical component = 4.5 km x sin(0°) = 0 km
For the second leg of the journey:
Horizontal component = 6 km x cos(32°)
Vertical component = 6 km x sin(32°)
Step 2: Add up the horizontal and vertical components.
Adding the horizontal components: 4.5 km + (6 km x cos(32°))
Adding the vertical components: 0 km + (6 km x sin(32°))
Step 3: Use the Pythagorean theorem to find the distance from the starting point.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
So, we can use the formula: distance^2 = (horizontal component)^2 + (vertical component)^2
Substituting the values from Step 2:
Distance^2 = (4.5 km + (6 km x cos(32°)))^2 + (0 km + (6 km x sin(32°)))^2
Simplifying and calculating the distance will give us the final answer.