# Math Help - word problem minimum distance

1. ## word problem minimum distance

this question is from my grade 12 calculus class I tried working it out but am having trouble drawing a correct diagram, which is the most important step for solving.

A power boat travels west at 24 km/h. At the instant it passes a buoy, a sailboat sailing north at 7 km/h is 25 km south of the buoy. Calculate the positions of the vessels when there is a minimum distance between them.

It would help if someone could help with the diagram to help get me started.

2. Originally Posted by math619
...
A power boat travels west at 24 km/h. At the instant it passes a buoy, a sailboat sailing north at 7 km/h is 25 km south of the buoy. Calculate the positions of the vessels when there is a minimum distance between them.
...
Hello,

I've attached a diagram as you requested.

Let d be the distance between the 2 boats.
Let t be the time the boats traveled since the powerboat passed the buoy.

Use Pythagorian rule:

$d^2=(24t)^2+(25-7t)^2$. Expand the brackets and collect equal powers:

$d^2=625t^2-350t+625$. d&#178; has a minimum if d has a minimum. Thus it is sufficient if you calculate the minimum of d&#178; = D

$D'(t)=1250t - 350$. D'(t) = 0 if $t=\frac{350}{1250}=\frac{7}{25}$

That means after 7/25 hours = 16 min 48 s, the distance is a minimum. Plugin this value for t and calculate d.

d = 24 km

EB