Question

Two boats are travelling at 30 miles/hr, the first going north and the second going east. The second crosses the path of the first 10 minutes after the first one was there. At what rate is their distance increasing when the second has gone 10 miles beyond the crossing point

Answer 1

their distance is increasing at the rate of 41.6 miles/hr

Explanation:

Given the data in the question;

first we determine the distance travelled by the first boat in 10 min when the second boat was crossing its path;

⇒ ( 30/60 ) × 10 = 5 miles

so as illustrated in the diagram below;

y² = x² + ( x + 5 )²

2y
`(dy)/(dt)` = 2x
`(dx)/(dt)` + 2(x+5)
`(dx)/(dt)`

y
`(dy)/(dt)` = ( 2x + 5 ) ]
`(dx)/(dt)`

`(dy)/(dt)` = [( 2x + 5 )/y ]
`(dx)/(dt)` ------ let this be equation 1

Now, given that,
`(dx)/(dt)` = 30 miles/hr, when x = 10

so

y = √( 10² + 15² ) = √325

so from equation 1

`(dy)/(dt)` = [( 2x + 5 )/y ]
`(dx)/(dt)`

we substitute

`(dy)/(dt)` = [( 2(10) + 5 ) / √325 ]30

`(dy)/(dt)` = [ 25 / √325 ] × 30

`(dy)/(dt)` = 41.6 miles/hr

Therefore, their distance is increasing at the rate of 41.6 miles/hr