Question

two people start biking from the same point. one heads east at 16 mph, the other south at 20 mph. what is the rate at which the distance between the two people is changing after 20 minutes and after 45 minutes? what is the rate at which the distance between the two people is changing after 20 minutes? rate at 20 minutes = 4√41/mi/hr what is the rate at which the distance between the two people is changing after 45 minutes? rate at 45 minutes =

Explanation:

Step1: Calculate the distance each person has traveled

• For the person going east: \(d_{east}=16\times t\) (where \(t\) is in hours).
• For the person going south: \(d_{south}=20\times t\).
• Using the Pythagorean theorem, the distance \(D\) between them is \(D = \sqrt{(16t)^{2}+(20t)^{2}}=\sqrt{256t^{2}+400t^{2}}=\sqrt{656t^{2}} = 4t\sqrt{41}\).
• The rate of change of \(D\) with respect to \(t\) (using the chain - rule, since \(D(t)\) and we want \(\frac{dD}{dt}\)), for \(D(t)=4\sqrt{41}t\), \(\frac{dD}{dt}=4\sqrt{41}\) (the rate is constant).

Step2: Confirm the rate at different times

Since \(\frac{dD}{dt}=4\sqrt{41}\) (a constant value, independent of \(t\) in the formula we derived for the rate of change of the distance between the two moving points).

Answer:

The rate at which the distance between the two people is changing after 45 minutes is \(4\sqrt{41}\frac{mi}{hr}\) (the same as the rate at 20 minutes because the rate of change of the distance between two objects moving at constant right - angled speeds is constant).