Algebra word problem help, please!

Question

Nicholas takes a boat ride down a river to visit an island at the mouth of the river. The boat leaves the docks at 9:15 am and reaches the island at 10:34 am. He spends several hours on the island, and leaves on the return boat at 5:45. Because the boat is moving upriver against the current, the return trip takes 67 minutes longer than the trip to the island. Let v be the cruising speed of the boat, c be the speed of the current, and d be the distance to the island.

 

Choose all the true statements:

 

1.) v-cv ≤ 0

 

2.) v/c=225/67

 

3.) dc/v^2-c^2-67/2

 

4.) d-79v/d-146v=79/146

Answer 1

Remember that distance = (rate)(time). In other words, (distance)/(time) = rate and (distance)/(rate) = time.

 

 

 

On the way to the island, the boat travels a distance of d. It travels with the current, so it travels at a rate of v + c. And it travels for a time of 79 minutes. So:

 

d = 79(v+c)

 

 

 

On the way back from the island, the boat travels the same distance of d. It travels against the current, so it travels at a rate of v - c. And it travels for a time of 79+67 = 146 minutes. So:

 

d = 146(v-c)

 

 

 

Combining the equations, we get 79(v+c) = 146(v-c).

 

Now, let's think about each statement one by one.

 

 

 

1) v - cv ≤ 0

 

This can be restated as v ≤ cv. Cruising speed and current will both be positive values, so cruising speed is less than or equal to (current)(cruising speed) when current is greater than or equal to 1. (That's because multiplying a number by 1 or more will leave it the same or increase it, while multiplying a number by a fraction will decrease it.)

 

So, this statement will be false when the current's speed is a fraction. Unfortunately, we're not told what units v and c should be expressed in. So if they're in, say, kilometers per second, the current is a fraction as long as it's moving at less than 2,236 miles per hour.

 

That means this statement isn't guaranteed to be true, but that's a fault of the problem, not the statement.

 

 

 

2) v/c = 225/67

 

Let's try to solve for v/c in our equation from above:

 

79(v+c) = 146(v-c)

79v + 79c = 146v - 146c

146c + 79c = 146v - 79v

225c = 67v

225 = 67v/c

225/67 = v/c

 

So, this statement is true.

 

 

 

3) dc/(v²-c²) = 67/2

 

The other tutor provided an explanation for this statement.

 

 

 

4) (d-79v)/(d-146v) = 79/146

 

This equation involves only d and v. So, let's solve our earlier equations for c and set them equal to each other:

 

d = 79(v+c)

d = 79v + 79c

d - 79v = 79c

(d - 79v)/79 = c

 

d = 146(v-c)

d = 146v - 146c

d - 146v = -146c

(d - 146v)/-146 = c

 

(d - 79v)/79 = (d - 146v)/-146

-146(d - 79v) = 79(d - 146v)

-146(d - 79v)/(d - 146v) = 79

(d - 79v)/(d - 146v) = -79/146

 

This is very close to the stated equation, but does not quite match. So, this statement is not true.

 

Answer 2

Given  speed of boat is =v

           speed of current is= c

          distance to island= d

          Time to travel to island= 10:34 - 9:15 = 79 minutes = d/ (v+c)

           Time to return  is 67 minutes more than going to island= 79+67 =d/ (v-c)

            d/(v-c) = d/(v+c)  +67

   least common denominator is (v-c)*(v+c)    

           d(v+c)  = d(v-c) + 67 (v^2 - c^2)

           dv+dc=  dv-dc  +67(v^2- c^2)

           2dc    = 67(v^2-c^2) 

         Divide both sides by 2

                dc = 67(v^2-c^2)/2

         Divide both sides by (v^2 - c^2)

         dc/(v^2- c^2) = 67/2     This is the statement at serial number 3 which is true