The distance between the ships is changing at a rate of 21.393 km/h
Step 1: First, you must draw an image to help better understand the problem
The reason that ship A is moving -35km/h is because ship B acts as an "origin" and as things move closer to the "origin" they become negative and as things move away from the "origin" they become positive.We know that the rate of Ship A is expressed as dAdt=−35 and the rate of Ship B is expressed as dBdt=25Step 2: We want to figure out how many km Ship A and Ship B traveled in 4 hoursShipA:−35kmh⋅4hr=−140kmNow if we take this −140km and add it to the 150km we can see that Ship A is now only 10km away from where Ship B beganShipB:25kmh⋅4hr=100kmThis means that Ship B is now 100km from where it originally startedWe can now redraw our informationStep 3: We must use the Pythagorean Theorem to find the third side of the trianglex2+y2=z2→102+1002=z2z=√102+1002Step 4: Now that we know z, we must differentiate the original equation in order to find the rate at which z in changingx2+y2=z2→2xdxdt+2ydydt=2zdzdtWe can simplify by canceling out the 2'sxdxdt+ydydt=zdzdtStep 5: Write down all the information and then plug into equationx=10anddxdt=−35
y=100anddydt=25
z=√102+1002anddzdt=?Now plug in the informationxdxdt+ydydt=zdzdt10(−35)+100(25)=√102+1002dzdt−350+2500=√102+1002dzdt2150=√102+1002dzdt21.393=dzdtThe distance between the ships are increasing at a rate of 21.393 km/h