A park has a circular walking path with a diameter of 250 meters.
a) Write the equation to describe the walking path in your diagram.
b) Find the distance traveled by someone walking once around the entire path.
c) Find the distance traveled by someone walking counterclockwise along the path from
the easternmost point of the path to the northernmost point of the path.
d) Find the distance traveled by someone walking straight through the park from the east-
ernmost point of the path to the northernmost point of the path.
well, you know that C = πd
The questions should be easy to answer, knowing that. Make diagrams if things aren't clear just from the words.
a) The equation to describe the walking path in the diagram can be written as:
(x - 0)^2 + (y - 0)^2 = (250/2)^2
This equation is in the form of a standard circle equation with a center at the origin (0, 0) and a radius of 125 meters (half of the diameter).
b) To find the distance traveled by someone walking once around the entire path, we need to calculate the circumference of the circle. The formula for the circumference of a circle is:
C = 2 * π * r
where C is the circumference, π is a constant equal to approximately 3.14159, and r is the radius. Substituting the given values:
C = 2 * 3.14159 * 125 = 785.398 meters
So, someone walking once around the entire path would travel a distance of approximately 785.398 meters.
c) To find the distance traveled by someone walking counterclockwise along the path from the easternmost point to the northernmost point, we need to calculate the length of an arc of the circle.
The central angle between the easternmost point and the northernmost point is 90 degrees. The formula to calculate the length of an arc is:
Length of arc = (angle in degrees / 360) * circumference
Substituting the given values:
Length of arc = (90 / 360) * 785.398 ≈ 196.35 meters
So, someone walking counterclockwise along the path from the easternmost point to the northernmost point would travel a distance of approximately 196.35 meters.
d) To find the distance traveled by someone walking straight through the park from the easternmost point to the northernmost point, we need to calculate the length of the straight line segment between these two points.
This line segment forms the hypotenuse of a right triangle where the radius (r) of the circle is the distance of one of the legs and the distance to be found is the hypotenuse.
Using the Pythagorean theorem, the formula to calculate the hypotenuse length is:
Hypotenuse = √(Leg1^2 + Leg2^2)
Substituting the given values:
Leg1 = Leg2 = 125 (since the radius is 125 meters)
Hypotenuse = √(125^2 + 125^2) = √(2 * 125^2) = √(2 * 15625) ≈ 176.78 meters
So, someone walking straight through the park from the easternmost point to the northernmost point would travel a distance of approximately 176.78 meters.
a) The equation to describe the walking path in the diagram can be written as:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the center of the circle and r represents the radius of the circle.
b) To find the distance traveled by someone walking once around the entire path, we need to find the circumference of the circle. The circumference of a circle is given by the formula:
C = 2πr
where r is the radius of the circle. In this case, the diameter (d) is given as 250 meters, so the radius (r) is half of the diameter. Therefore, the radius is:
r = d/2 = 250/2 = 125 meters
Now, we can calculate the circumference:
C = 2π(125) = 250π meters
So, the distance traveled by someone walking once around the entire path is 250π meters.
c) To find the distance traveled by someone walking counterclockwise along the path from the easternmost point of the path to the northernmost point of the path, we need to find the length of the arc between these two points.
The angle between the easternmost point and the northernmost point on the circle is 90 degrees (π/2 radians) since the path is circular.
The formula to calculate the length (L) of an arc of a circle is given by:
L = (θ/360) * 2πr
where θ is the central angle (in radians) formed by the two points on the circle.
In this case, the central angle θ is 90 degrees (or π/2 radians). The radius (r) is 125 meters. Plugging these values into the formula, we get:
L = (π/2/360) * 2π(125) = (π/360) * 2π(125) = (π/360) * 250π = 250/360 * π^2 meters
So, the distance traveled by someone walking counterclockwise along the path from the easternmost point to the northernmost point is (250/360)π^2 meters.
d) To find the distance traveled by someone walking straight through the park from the easternmost point of the path to the northernmost point of the path, we need to find the length of the straight line segment connecting these two points.
The length of a straight line segment connecting two points can be calculated using the distance formula:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
In this case, the easternmost point has coordinates (125, 0) and the northernmost point has coordinates (0, 125) (assuming the center of the circle is at (0, 0)). Plugging these values into the distance formula, we get:
d = √[(0 - 125)^2 + (125 - 0)^2] = √[(-125)^2 + 125^2] = √[15625 + 15625] = √31250 meters
So, the distance traveled by someone walking straight through the park from the easternmost point of the path to the northernmost point of the path is √31250 meters.