Let's solve the problem step by step.
First, let's define the given information:
- The motorboat takes 5 hours to travel 150 miles upstream.
- The same boat takes 3 hours to travel 150 miles downstream.
### Step 1: Calculate the speed upstream
The speed of the boat going upstream can be found using the formula:
[tex]\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \][/tex]
For the upstream trip:
[tex]\[ \text{Speed upstream} = \frac{150 \text{ miles}}{5 \text{ hours}} = 30 \frac{\text{mi}}{\text{h}} \][/tex]
### Step 2: Calculate the speed downstream
Similarly, the speed of the boat going downstream is:
[tex]\[ \text{Speed downstream} = \frac{150 \text{ miles}}{3 \text{ hours}} = 50 \frac{\text{mi}}{\text{h}} \][/tex]
### Step 3: Calculate the rate of the boat in still water
The rate of the boat in still water is the average of the upstream and downstream speeds. This is because the effect of the current cancels out when averaging.
[tex]\[ \text{Rate of the boat in still water} = \frac{\text{Speed upstream} + \text{Speed downstream}}{2} \][/tex]
Substituting the values we have:
[tex]\[ \text{Rate of the boat in still water} = \frac{30 \frac{\text{mi}}{\text{h}} + 50 \frac{\text{mi}}{\text{h}}}{2} = \frac{80 \frac{\text{mi}}{\text{h}}}{2} = 40 \frac{\text{mi}}{\text{h}} \][/tex]
### Step 4: Calculate the rate of the current
The rate of the current can be found as half the difference between the downstream and upstream speeds, as the current adds to the boat speed in one direction and subtracts from it in the other direction.
[tex]\[ \text{Rate of the current} = \frac{\text{Speed downstream} - \text{Speed upstream}}{2} \][/tex]
Substituting the values we have:
[tex]\[ \text{Rate of the current} = \frac{50 \frac{\text{mi}}{\text{h}} - 30 \frac{\text{mi}}{\text{h}}}{2} = \frac{20 \frac{\text{mi}}{\text{h}}}{2} = 10 \frac{\text{mi}}{\text{h}} \][/tex]
### Final Answers
- Rate of the boat in still water: [tex]\( 40 \frac{\text{mi}}{\text{h}} \)[/tex]
- Rate of the current: [tex]\( 10 \frac{\text{mi}}{\text{h}} \)[/tex]
