Orville walks 330 m due east. He then continues walking along a straight line, but in a different direction, and stops 160 m northeast of his starting point. How far did he walk during the second portion of the trip and in what direction?
use the law of cosines. The distance z is found using
z^2 = 330^2 + 160^2 - 2*330*160 cos 45°
and the direction is θ (measured from due east) where
tanθ = (160/√2)/(160/√2 - 330)
To find the distance that Orville walked during the second portion of the trip, we can use the Pythagorean theorem.
Let's define the distance walked during the second portion as "d" and the angle between the initial east direction and the second direction as "θ".
Using the given information, we can set up the following equation:
(330m)^2 + d^2 = (160m)^2
Simplifying the equation, we have:
108900 + d^2 = 25600
Subtracting 25600 from both sides:
d^2 = 83250
Taking the square root of both sides:
d ≈ √83250
Calculating the square root, we get:
d ≈ 288.64 m
Therefore, Orville walked approximately 288.64 m during the second portion of the trip.
To determine the direction, we can use trigonometry. Since Orville walked northeast, we can calculate the angle θ using the tangent function:
tan(θ) = (opposite/adjacent) = (160m/330m)
θ ≈ tan^(-1)(160/330)
θ ≈ 25.43 degrees (rounded to two decimal places)
Therefore, Orville walked approximately 288.64 m in a direction approximately 25.43 degrees northeast.
To solve this problem, we can break it down into two components: the distance walked and the direction of the second portion of the trip.
First, let's find the distance walked during the second portion of the trip. We know that Orville walked 330 m due east initially, and then stopped 160 m northeast of his starting point.
To find the distance walked during the second portion, we can calculate the difference between the total distance traveled and the distance walked initially.
Total distance traveled = distance walked initially + distance walked during second portion
Given that Orville walked 330 m due east initially, we need to find the distance walked during the second portion.
Let's assume the distance walked during the second portion is "d".
Therefore, using the information given:
Total distance traveled = 330 m + d
We also know that Orville stopped 160 m northeast of his starting point. Northeast is a combination of 45 degrees east and 45 degrees north.
So, using trigonometry, we can determine the distance walked during the second portion:
d = 160 / cos(45)
Let's calculate the value of "d":
d = 160 / cos(45)
d ≈ 160 / 0.707
d ≈ 226.28 m
Therefore, Orville walked approximately 226.28 m during the second portion of the trip.
Now, let's determine the direction. We know that northeast is a combination of 45 degrees east and 45 degrees north.
So, to find the direction of the second portion, we can calculate the angle between the east direction and the direction in which Orville stopped.
Using trigonometry, we can find this angle:
Angle = arctan(160 / 330)
Let's calculate the value of the angle:
Angle = arctan(160 / 330)
Angle ≈ 24.69 degrees
Therefore, Orville walked approximately 226.28 m in a direction that is approximately 24.69 degrees north of east during the second portion of the trip.