A sailboat is 12 km north of a lighthouse. A motor cruiser is 12 km east of the same lighthouse.
- Use trigonometry to find an exact expression for the distance tween the two boats.
- Check your answer using another method.
nvm I got it
in math text coordinates:
A is at ( 0 , 12 )
B is at ( 12 , 0 )
draw the line between. It is the hypotenuse of a right triangle.
d = sqrt ( 12^2 + 12^2) = 12 sqrt 2
to sail from A to B I must go on a vector
V = +12 i - 12 j = 12 (i - j)
the magnitude of 1i -1j is sqrt 2
To find the exact expression for the distance between the two boats using trigonometry, we can analyze the problem as a right triangle.
Let's label the sailboat as point S, the lighthouse as point L, and the motor cruiser as point C.
From the given information, we have:
- S is 12 km north of L.
- C is 12 km east of L.
Now, we can form a right triangle with S, L, and C as the vertices.
We know the side adjacent to the angle between S and C (the distance between S and C) is 12 km. The side opposite to the angle (the distance between S and L) is also 12 km.
By applying the Pythagorean theorem, we can determine the hypotenuse, which represents the distance between the sailboat and the motor cruiser:
Distance between S and C = √(Distance between S and L)^2 + (Distance between C and L)^2
= √(12 km)^2 + (12 km)^2
= √(144 km^2 + 144 km^2)
= √(288 km^2)
= √(288) km
≈ 16.970 km
Therefore, using trigonometry, the exact expression for the distance between the sailboat and the motor cruiser is approximately 16.970 km.
To check this answer using another method, we can use the distance formula:
Distance = √((Change in x)^2 + (Change in y)^2)
From the given information, we know that the sailboat is 12 km north of the lighthouse, and the motor cruiser is 12 km east of the same lighthouse.
The change in x between S and C is 12 km, and the change in y between S and C is 12 km.
Plugging these values into the distance formula, we get:
Distance = √((12 km)^2 + (12 km)^2)
= √(144 km^2 + 144 km^2)
= √(288 km^2)
≈ 16.970 km
This again confirms that the distance between the sailboat and the motor cruiser is approximately 16.970 km.