Two people start from the same point. One walks east at 4 mi/h and the other walks northeast at 3 mi/h. How fast is the distance between the people changing after 15 minutes? (Round your answer to three decimal places.)
0.6 of students walk to school. Of those 3/4 live within 1 mile of the school there are 640 students at the school. How many students walk to school and live 1 mile or farther from school?
Thanks for the Help!
To answer this question, we can use the concept of related rates. We want to find how fast the distance between the two people is changing, so we need to find the rate of change of the distance with respect to time.
Let's start by assigning variables to the relevant quantities:
- Let x be the distance traveled by the person walking east.
- Let y be the distance traveled by the person walking northeast.
- Let d be the distance between the two people.
Now, we can set up equations based on the given information:
- The person walking east travels at a rate of 4 mi/h, so x = 4t, where t is in hours.
- The person walking northeast travels at a rate of 3 mi/h, so y = 3t.
- The distance between the two people is given by the Pythagorean theorem: d = sqrt(x^2 + y^2).
We want to find the rate of change of d with respect to time (t), specifically after 15 minutes. So, we need to differentiate the equation for d with respect to t.
Differentiating both sides of the equation d = sqrt(x^2 + y^2) with respect to t using the chain rule, we get:
(d/dt) d = (d/dt) sqrt(x^2 + y^2)
Now, let's substitute the given information and solve the problem.
We know x = 4t and y = 3t, so we can substitute these expressions into the equation for d:
d = sqrt((4t)^2 + (3t)^2)
Simplifying further, we have:
d = sqrt(16t^2 + 9t^2)
d = sqrt(25t^2)
d = 5t
Now, let's differentiate both sides of this equation with respect to t:
(d/dt) d = (d/dt) (5t)
The left side represents the rate of change we want to find, and the right side is simply 5.
Therefore, we have:
d/dt = 5
The derivative of d with respect to t is a constant value of 5, indicating that the distance between the two people is changing at a constant rate of 5 mi/h.
To find the rate of change after 15 minutes (0.25 hours), we substitute t = 0.25 into the derivative:
d/dt = 5 * (t = 0.25)
d/dt = 5 * 0.25
d/dt = 1.25
Hence, the rate at which the distance between the two people is changing after 15 minutes is 1.25 miles per hour.