The distance between the ships changes at 7 P.M. by approximately 21.4 kilometers per hour.
Let suppose that both ships travel at constant speed. After a quick reading of the statement, we find that distance between the ships ([tex]r[/tex]), in kilometers, as a function of time ([tex]t[/tex]), in hours, is defined by Pythagorean theorem:
[tex]r = \sqrt{(r_{o}-\dot x \cdot t)^{2}+(\dot y\cdot t)^{2}}[/tex] (1)
Where:
Now we proceed to find an expression for the rate of change of the distance between the ships ([tex]\dot r[/tex]), in kilometers per hour:
[tex]\dot r = \frac{[(r_{o}-\dot x\cdot t)\cdot (-\dot x)+(\dot y\cdot t)\cdot \dot y]}{ \sqrt{(r_{o}-\dot x\cdot t)^{2}+(\dot y\cdot t)^{2}}}[/tex] (2)
If we know that [tex]r_{o} = 150\,km[/tex], [tex]\dot x = 35\,\frac{km}{h}[/tex], [tex]\dot y = 25\,\frac{km}{h}[/tex] and [tex]t = 4\,h[/tex], then the rate of change between the ships is:
[tex]\dot r = \frac{\left\{\left[150\,km-\left(35\,\frac{km}{h}\right)\cdot (4\,h)\right]\cdot \left(-35\,\frac{km}{h} \right)+\left[\left(25\,\frac{km}{h}\right)\cdot (4\,h)\right]\cdot \left(25\,\frac{km}{h} \right)\right\}}{\sqrt{\left[150\,km-\left(35\,\frac{km}{h}\right)\cdot (4\,h)\right]^{2}+\left[\left(25\,\frac{km}{h}\right)\cdot (4\,h)\right]^{2}}}[/tex]
[tex]\dot r \approx 21.393\,\frac{km}{h}[/tex]
The distance between the ships changes at 7 P.M. by approximately 21.4 kilometers per hour. [tex]\blacksquare[/tex]
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