Two people start from the same point. One walks east at 5 mi/h and the other walks northeast at 7 mi/h. How fast is the distance between the people changing after 15 minutes? (Round your answer to three decimal places.)
At time t, using the law of cosines,
d^2 = (7t)^2 + (5t)^2 - 2(5t)(7t)cosπ/4
= 74t^2 - 35√2 t^2
= (74-35√2)t^2
so,
2d dd/dt = 2(74-35√2)t
Now, plug in your numbers to get d(15), and solve for dd/dt at t=15
Remember to use appropriate units
17
Two people start from the same point. One walks east at 6 mi/h and the other walks northeast at 8 mi/h. How fast is the distance between the people changing after 15 minutes? (Round your answer to three decimal places.)
Well, since both people started from the same point and are walking away from each other, I'd say the distance between them is changing at the speed of the Usain Bolt chasing after a banana peel. Just kidding! Let's calculate it for real, shall we?
After 15 minutes, which is 1/4th of an hour, the eastward walker will have covered a distance of 5 * 1/4 = 1.25 miles.
Meanwhile, the walker moving northeast will have covered a distance of 7 * 1/4 = 1.75 miles.
So, the distance between them will be decreasing at a rate of 1.75 - 1.25 = 0.5 miles per 15 minutes. However, we need to convert that to miles per hour, so let me put on my math shoes.
Since there are four 15-minute intervals in an hour, we can say that the rate of change is 0.5 * 4 = 2 miles per hour.
So, the distance between them is changing at 2 miles per hour after 15 minutes. That's enough time to walk off some energy and get our calculations right!
To find the rate at which the distance between the two people is changing, we can use the concept of relative motion.
Let's first visualize the scenario.
We have two people starting from the same point. One person walks directly east at a speed of 5 mi/h, while the other person walks northeast at a speed of 7 mi/h.
After 15 minutes, the person walking east will have covered a distance of (5 mi/h) * (15/60) h = 1.25 miles.
To find the rate at which the distance between the two people is changing after 15 minutes, we need to determine how the person walking northeast affects this distance.
Since one person is moving east and the other person is moving northeast, we can form a right triangle using the distance traveled by the person walking east, the distance traveled by the person walking northeast, and the distance between them.
The distance between the two people is given by the Pythagorean theorem:
distance^2 = (1.25 miles)^2 + (distance traveled by the person walking northeast)^2
Rearranging the equation, we have:
(distance traveled by the person walking northeast)^2 = distance^2 - (1.25 miles)^2
To find the rate at which the distance between the two people is changing, we need to differentiate both sides of the equation with respect to time.
Differentiating, we have:
2 * (distance traveled by the person walking northeast) * (rate at which distance is changing) = 2 * distance * (rate at which distance between people is changing)
Now, let's plug in the values we know.
At the given time, the distance between the two people is the hypotenuse of the right triangle, which we'll call "d".
(d)^2 = (1.25 miles)^2 + (distance traveled by the person walking northeast)^2
Plugging in the known values, we have:
(d)^2 = (1.25 miles)^2 + (7 mi/h)^2 * (15/60) h)^2
Simplifying, we have:
(d)^2 = 1.5625 miles^2 + (1.225 miles)^2
(d)^2 = 1.5625 miles^2 + 1.495625 miles^2
(d)^2 = 3.058125 miles^2
Taking the square root of both sides, we have:
d = sqrt(3.058125) miles
Now we substitute this solution back into our derived equation:
2 * (distance traveled by the person walking northeast) * (rate at which distance is changing) = 2 * distance * (rate at which distance between people is changing)
2 * (7 mi/h) * (rate at which distance is changing) = 2 * sqrt(3.058125) miles * (rate at which distance between people is changing)
Simplifying further, we have:
14 mi/h * (rate at which distance is changing) = 2 * sqrt(3.058125) * (rate at which distance between people is changing)
Dividing both sides by 14 mi/h:
(rate at which distance is changing) = (sqrt(3.058125) / 7) * (rate at which distance between people is changing)
To find the rate at which the distance between the people is changing after 15 minutes, we need to find the rate at which the distance between the people is changing.
Finally, we calculate the rate at which the distance between the people is changing using the given rates:
(rate at which distance is changing) = (sqrt(3.058125) / 7) * (7 mi/h)
(rate at which distance is changing) = sqrt(3.058125) miles/h
Calculating the value, rounding to three decimal places, we get:
(rate at which distance is changing) ≈ 1.747 miles/h
Therefore, the rate at which the distance between the two people is changing after 15 minutes is approximately 1.747 miles/h.