# word problem minimum distance

• Jan 5th 2007, 07:01 AM
math619
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.
• Jan 5th 2007, 10:29 AM
earboth
Quote:

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:

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

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

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

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