Question 1089509
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<U>Answer</U>. The rate of the cruise ship for the first &nbsp;40 mi&nbsp; is &nbsp;10 miles per hour.


<U>Check</U>. &nbsp;&nbsp;{{{40/10 + 60/(10+5)}}} = 4 + 4 = 8 hours.   &nbsp;&nbsp;! Correct !


<B>Solution</B>


<pre>
Let "r" be the slower rate of the cruise ship, in miles per hour.


Then the faster speed is (r+5) miles per hour.


The time spent to cover 40 miles at the rate "r" mph is {{{40/r}}} hours.

The time spent to cover 60 miles at the rate (r+5) mph is {{{60/(r+5)}}} hours.

The condition says that the total time is 8 hours:

{{{40/r}}} + {{{60/(r+5)}}} = 8.


To solve this equation, multiply both sides by r*(r+5). You will get

40*(r+5) + 60*r = 8r*(r+5).


Simplify and solve for "r".

40r + 200 + 60r = 8r^2 + 40r,

8r^2 - 60r - 200 = 0,   ====>  (divide both sides by 4)  ====>

2r^2 - 15r - 50 = 0,

{{{r[1,2]}}} = {{{(15 +- sqrt(15^2 -4*2*(-50)))/(2*2)}}} = {{{(15 +- 25)/4}}}.


Only positive root r= 10 works.


<U>Answer</U>. The rate of the cruise ship for the first &nbsp;40 mi&nbsp; is &nbsp;10 miles per hour.
</pre>

Solved.



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I edited my post.


I apologize for my words in previous version.