Virgil sets his boat on a 1000-yard course keeping a constant distance from a rocky outcrop.
If Virgil keeps a distance of 200 yards, through what angle does he travel?
If Virgil keeps a distance of 500 yards, what fraction of the circumference of a circle does he cover?
360 degrees around = 2 pi r = 400 pi = 1257 yards
(1000/1257)360 = 286 degrees
tset
To answer the first question, we can draw a right triangle to represent the situation. Let's call the distance between Virgil's boat and the rocky outcrop 'd', and the 1000-yard course 'c'. We know that d = 200 yards.
Now, by drawing these lengths as two sides of a right triangle and the 1000-yard course as the hypotenuse, we can use trigonometry to find the angle. The relationship we can use is the tangent function.
The tangent of an angle is defined as the ratio of the opposite side (d) to the adjacent side (c). So, we have:
tangent(angle) = d / c
Substituting the values, we get:
tangent(angle) = 200 / 1000
To find the angle, we need to find the inverse tangent (also called arctan) of both sides:
angle = arctan(200 / 1000)
Calculating this, we find:
angle ≈ 11.31 degrees
Therefore, Virgil travels at an angle of approximately 11.31 degrees.
Moving on to the second question, if Virgil keeps a distance of 500 yards, we can consider the fraction of the circumference of a circle he covers.
To find this fraction, we need to determine the length of the arc covered by Virgil's boat. The formula to calculate the length of an arc is:
Arc length = (angle / 360) * (2 * pi * r)
Here, 'angle' represents the angle (in degrees) covered by Virgil's boat, and 'r' represents the distance between Virgil's boat and the rocky outcrop, which is 500 yards.
First, let's calculate the angle (in degrees) using the same tangent relationship as before:
tangent(angle) = d / c
angle = arctan(500 / 1000)
Calculating this, we find:
angle ≈ 26.57 degrees
Now, we can substitute the values into the arc length formula:
Arc length = (26.57 / 360) * (2 * pi * 500)
Calculating this, we get:
Arc length ≈ 229.18 yards
To find the fraction of the circumference covered, we divide the arc length by the circumference of the entire circle, which is 2 * pi * 500:
Fraction covered = Arc length / (2 * pi * 500)
Calculating this fraction, we have:
Fraction covered ≈ 229.18 / (2 * pi * 500)
Fraction covered ≈ 0.0729
Therefore, if Virgil keeps a distance of 500 yards, he covers approximately 0.0729 or 7.29% of the circumference of a circle.
To find the answers to these questions, we can use some trigonometry and geometry concepts.
1. If Virgil keeps a distance of 200 yards, we need to calculate the angle he travels. Let's call the distance between Virgil's boat and the rocky outcrop "d."
To find the angle, we can use the inverse cosine function (arccos) since we know the adjacent side (200 yards) and the hypotenuse (1000 yards). The relationship can be expressed as:
cos(angle) = adjacent / hypotenuse
Plugging in the values, we get:
cos(angle) = 200 / 1000
Simplifying further:
cos(angle) = 1/5
Now, we can find the angle by taking the inverse cosine of both sides:
angle = arccos(1/5)
Using a calculator or reference table, we can find the angle to be approximately 78.46 degrees.
Therefore, Virgil travels at an angle of approximately 78.46 degrees when maintaining a distance of 200 yards from the rocky outcrop.
2. If Virgil keeps a distance of 500 yards, we want to determine the fraction of the circumference of a circle that he covers.
To calculate this fraction, we first need to find the length of the circle's circumference. The formula for the circumference of a circle is given by:
Circumference = 2 * pi * radius
Given that the distance Virgil keeps from the rocky outcrop (500 yards) is the radius of the circle, we can plug this value into the formula:
Circumference = 2 * pi * 500
Simplifying further:
Circumference = 1000 * pi
Now, to find the fraction of the circumference that Virgil covers, we need to determine the length of the arc that corresponds to the distance he traveled. The length of an arc can be calculated using the formula:
Arc Length = (angle / 360 degrees) * Circumference
We already calculated the angle Virgil travels in the first part, which was approximately 78.46 degrees. Plugging this value into the formula, we get:
Arc Length = (78.46 / 360) * (1000 * pi)
Simplifying further:
Arc Length = (0.2179) * (1000 * pi)
Arc Length ≈ 217.9 pi
To find the fraction of the circumference, we can divide the arc length by the circumference:
Fraction = Arc Length / Circumference
Plugging in the values, we get:
Fraction ≈ (217.9 pi) / (1000 pi)
Simplifying further:
Fraction ≈ 0.2179
Therefore, when Virgil keeps a distance of 500 yards, he covers approximately 0.2179 (or 21.79%) of the circumference of the circle.