the shoreline of circle lake is a circle with diameter 2 mi.nancy's training routine begins at point e on the eastern shore of the lake.she jogs along the north shore to a point p and then swims the straight line distance,if any point ,from point to point w diametrically opposite.nancy swims at a rate of 2 mi/h and josh 8 mi/h how far should nancy jog in order to complete her training routine?what is the least and greatest amount of time answer?
Label the center of the circle O. Circle O has radius 1.
If angle EOP is θ, then she
jogs a distance of rθ = θ
swims a distance of r sin(π-θ)/2 = r sin(π/2-θ/2) = r cos(θ/2) = cos θ/2
so, since time = distance/speed, her time is
t(θ) = θ/8 + 1/2 cos(θ/2)
To find min/max, set dt/θ = 0 and solve for θ.
Well, you see, Nancy is jogging and swimming, while Josh is just chilling at a constant speed. Poor Josh, he must be tired of waiting for Nancy.
So, let's figure this out. The circle's diameter is 2 miles, which means the radius is 1 mile. Nancy starts at point E, so she needs to jog to point P on the north shore.
The length of the circumference of the circle can be found using the formula C = 2πr, where r is the radius. In our case, C = 2π(1) = 2π miles. Now, we divide this circumference by 2 to find the distance from E to P. That gives us π miles.
Now, we need to figure out the time it will take Nancy to swim from P to the diametrically opposite point W. Since Nancy swims at 2 mi/h, the time is simply the distance divided by her swimming speed. So, t = π miles / 2 mi/h = π/2 hours.
As for the least and greatest amount of time, well, that depends on Nancy's jogging speed. Let's say she runs x miles per hour. The time it will take her to jog from E to P is simply the distance divided by her jogging speed, which is π miles / x mi/h = π/x hours.
So, the total time for her routine is t + π/x hours.
Now, to minimize the time, Nancy should jog at her fastest speed. On the other hand, to maximize the time, she should jog at her slowest speed. But keep in mind, poor Josh will have to wait all that time!
I hope that answers your question, or at least confuses you enough to forget about the actual numbers. Stay clownish!
To complete her training routine, Nancy needs to jog along the north shore and then swim the straight line distance from point P to point W, which is diametrically opposite.
Given that the diameter of Circle Lake is 2 miles, the radius is half of that, which is 1 mile.
To find the distance Nancy should jog, we need to find the length of Arc NP.
The circumference of a circle is calculated using the formula C = 2πr, where r is the radius, and π is a constant approximately equal to 3.14.
Therefore, the circumference of Circle Lake is C = 2π(1) = 2π miles.
To find the length of Arc NP, we need to know the angle that it subtends at the center of the circle. Since this information is not provided, we cannot determine the exact length of Arc NP.
However, we can find the least and greatest possible distances that Nancy can jog by considering extreme cases.
Least Distance:
If Nancy runs along the shortest route from point E to point P, she will jog along the semicircle. In this case, Arc NP will be half of the circumference of Circle Lake.
Arc NP = (1/2)(2π) = π miles
Greatest Distance:
If Nancy runs along the longest route from point E to point P, she will jog along the full circle. In this case, Arc NP will be equal to the circumference of Circle Lake.
Arc NP = 2π miles
To find the least and greatest amount of time, we need to calculate the time it takes for Nancy to swim from point P to point W.
The swimming speed of Nancy is given as 2 mi/h. Therefore, it will take her (Arc NP) / (2 mi/h) hours to swim this distance.
Least Time:
Time taken to swim the least distance = π / 2 hours
Greatest Time:
Time taken to swim the greatest distance = 2π / 2 hours = π hours
Therefore, the least and greatest amount of time required for Nancy to complete her training routine are π / 2 hours and π hours, respectively.
To determine how far Nancy should jog in order to complete her training routine, we need to understand the given information about Circle Lake and the activities involved.
The shoreline of Circle Lake is a circle with a diameter of 2 miles. This means that the circumference of the lake is π times the diameter, i.e., π x 2 miles = 2π miles.
Nancy's training routine begins at point E on the eastern shore of the lake. She jogs along the north shore to point P.
To find the distance Nancy should jog, we need to find the arc length from point E to point P on the northern part of the lake.
The length of an arc on a circle can be found using the formula:
Arc Length = (θ/360) * 2πr
where θ is the central angle in degrees and r is the radius of the circle.
In this case, since the distance from E to P is along the shoreline, it can be considered as an arc on the circle with the central angle as 90 degrees.
Since the circle's diameter is 2 miles, the radius is 1 mile. Therefore, the arc length Nancy should jog is:
Arc Length = (90/360) * 2π * 1 mile
= (1/4) * 2π mile
= 0.5π mile
Now, we know that Nancy swims the straight-line distance from any point on the circle to the diametrically opposite point.
To find the diametrically opposite point of P, we divide the circle into two equal halves. So, the diametrically opposite point of P will be the southernmost point on the shore of the lake.
Since Nancy swims at a rate of 2 miles per hour, we can calculate the time it would take for her to swim from P to the opposite point:
Time = Distance / Speed
= 2 miles / 2 miles per hour
= 1 hour
Therefore, the least amount of time Nancy would take to complete her training routine is the time it takes for her to swim from P to the diametrically opposite point, which is 1 hour.
The greatest amount of time Nancy would take is the time it would take for her to jog the entire circumference of the lake (2π miles) and then swim back to the starting point from the diametrically opposite point. Let's calculate it.
Time taken to jog the entire circumference of the lake = Distance / Speed
= 2π miles / 8 miles per hour
= 0.25π hour
Time taken to swim back from the diametrically opposite point = 1 hour
The greatest amount of time Nancy would take = 0.25π hour + 1 hour = (0.25π + 1) hours.
So, Nancy should jog approximately 0.5π miles to complete her training routine, and the least amount of time is 1 hour, while the greatest amount of time is (0.25π + 1) hours.