Jasper walks 2 km due east. He then turns and walks 3 km northeast.
a. How far is Jasper from his starting point? Round your answer to 2 decimal places.
b. What is the direction of Jasper's destination in relation to his starting point? Round your answer to the nearest degree.
Did you make a diagram?
I see a triangle with sides 2 and 3 and the contained angle as 135º, let the side opposite that angle be x
I see the cosine law.
x^2 = 2^2 + 3^2 - 2(2)(3)cos 135
= 4 + 9 - 12(-√3/2)
= ....
x = √....
then use the sine law to find the angle at the origin
sin A/3 = sin 135/x
....
To find the distance from Jasper's starting point (a), we can use the Pythagorean theorem. 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.
In this case, Jasper walks 2 km due east (which we can consider as the "x" direction) and then turns and walks 3 km northeast (which forms a 45-degree angle with the positive x-axis). This creates a right triangle.
Step 1: Calculate the horizontal distance Jasper has traveled (Δx) by walking 2 km due east.
Δx = 2 km
Step 2: Calculate the vertical distance Jasper has traveled (Δy) by walking 3 km northeast.
Since it forms a 45-degree angle with the positive x-axis, the vertical distance is equal to the horizontal distance.
Δy = 2 km
Step 3: Calculate the distance from Jasper's starting point using the Pythagorean theorem.
Distance = √(Δx^2 + Δy^2)
Distance = √((2 km)^2 + (2 km)^2)
Distance = √(4 km^2 + 4 km^2)
Distance = √(8 km^2)
Distance ≈ 2.83 km (rounded to 2 decimal places)
So, Jasper is approximately 2.83 km from his starting point.
To find the direction of Jasper's destination in relation to his starting point (b), we can use trigonometry.
Step 1: Calculate the angle Jasper forms with the positive x-axis.
Angle = arctan(Δy / Δx)
Angle = arctan(2 km / 2 km)
Angle = arctan(1)
Angle ≈ 45 degrees (rounded to the nearest degree)
So, the direction of Jasper's destination in relation to his starting point is approximately 45 degrees.