## To solve for the time at which P and Q are closest to each other, you can use the distance formula.

The distance, D, between two points (x1, y1) and (x2, y2) is given by:

D = sqrt((x2 - x1)^2 + (y2 - y1)^2)

In this case, let's consider the positions of P and Q at time t:

Position of P: (10 - 30t, 0)

Position of Q: (0, 10 - 40t)

Now, substitute these values into the distance formula:

D = sqrt((0 - (10 - 30t))^2 + ((10 - 40t) - 0)^2)

= sqrt((30t - 10)^2 + (10 - 40t)^2)

= sqrt(900t^2 - 600t + 100 + 100t^2 - 800t + 100)

= sqrt(1000t^2 - 1400t + 200)

To find the time at which P and Q are closest to each other, we need to minimize this distance function. Take the derivative of D with respect to t:

D' = (1/2)(1000t^2 - 1400t + 200)^(-1/2)(2000t - 1400)

Set D' equal to 0 and solve for t:

(1/2)(1000t^2 - 1400t + 200)^(-1/2)(2000t - 1400) = 0

Simplifying this equation, we get:

2000t - 1400 = 0

t = 7/10

Therefore, at t = 7/10, P and Q are closest to each other.