<span>The distance between the ships is changing at a rate of 21.393 km/h
</span><span>Step 1: First, you must draw an image to help better understand the problem
</span>
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 <span><span><span>dA</span><span>dt</span></span>=−35</span> and the rate of Ship B is expressed as <span><span><span>dB</span><span>dt</span></span>=25</span>
Step 2: We want to figure out how many km Ship A and Ship B traveled in 4 hours
<span>ShipA:−35<span><span>km</span>h</span>⋅4hr=−140km</span>
Now if we take this <span>−140km</span> and add it to the <span>150km</span> we can see that Ship A is now only <span>10km</span> away from where Ship B began
<span>ShipB:25<span><span>km</span>h</span>⋅4hr=100km</span>
This means that Ship B is now <span>100km</span> from where it originally started
We can now redraw our information
Step 3: We must use the Pythagorean Theorem to find the third side of the triangle
<span><span>x2</span>+<span>y2</span>=<span>z2</span>→<span>102</span>+<span>1002</span>=<span>z2</span></span>
<span>z=<span>√<span><span>102</span>+<span>1002</span></span></span></span>
Step 4: Now that we know z, we must differentiate the original equation in order to find the rate at which z in changing
<span><span>x2</span>+<span>y2</span>=<span>z2</span>→2x<span><span>dx</span><span>dt</span></span>+2y<span><span>dy</span><span>dt</span></span>=2z<span><span>dz</span><span>dt</span></span></span>
We can simplify by canceling out the 2's
<span>x<span><span>dx</span><span>dt</span></span>+y<span><span>dy</span><span>dt</span></span>=z<span><span>dz</span><span>dt</span></span></span>
Step 5: Write down all the information and then plug into equation
<span>x=10and<span><span>dx</span><span>dt</span></span>=−35</span>
<span>y=100and<span><span>dy</span><span>dt</span></span>=25</span>
<span>z=<span>√<span><span>102</span>+<span>1002</span></span></span>and<span><span>dz</span><span>dt</span></span>=?</span>
Now plug in the information
<span>x<span><span>dx</span><span>dt</span></span>+y<span><span>dy</span><span>dt</span></span>=z<span><span>dz</span><span>dt</span></span></span>
<span>10<span>(−35)</span>+100<span>(25)</span>=<span>√<span><span>102</span>+<span>1002</span></span></span><span><span>dz</span><span>dt</span></span></span>
<span>−350+2500=<span>√<span><span>102</span>+<span>1002</span></span></span><span><span>dz</span><span>dt</span></span></span>
<span>2150=<span>√<span><span>102</span>+<span>1002</span></span></span><span><span>dz</span><span>dt</span></span></span>
<span>21.393=<span><span>dz</span><span>dt</span></span></span>
The distance between the ships are increasing at a rate of 21.393 km/h
