Let's address Question 15 step by step to determine the speed of the boat in still water and the speed of the current.
1. Identify Given Information:
- Distance traveled downstream: 50 miles
- Time taken to travel downstream: 2 hours
- Time taken to travel upstream: 5 hours
2. Define Variables:
- Let [tex]\( b \)[/tex] be the speed of the boat in still water (in miles per hour).
- Let [tex]\( c \)[/tex] be the speed of the current (in miles per hour).
3. Determine Downstream and Upstream Speeds:
- Downstream speed (speed of boat with current): [tex]\( b + c \)[/tex]
- Upstream speed (speed of boat against current): [tex]\( b - c \)[/tex]
4. Establish Equations:
- Using the formula [tex]\( \text{speed} = \frac{\text{distance}}{\text{time}} \)[/tex]:
- Downstream speed:
[tex]\[
b + c = \frac{50}{2} = 25 \text{ miles per hour}
\][/tex]
- Upstream speed:
[tex]\[
b - c = \frac{50}{5} = 10 \text{ miles per hour}
\][/tex]
5. Solve the System of Equations:
- We now have two linear equations:
[tex]\[
\begin{cases}
b + c = 25 \\
b - c = 10
\end{cases}
\][/tex]
- Add the two equations to eliminate [tex]\( c \)[/tex]:
[tex]\[
(b + c) + (b - c) = 25 + 10
\][/tex]
[tex]\[
2b = 35
\][/tex]
[tex]\[
b = \frac{35}{2} = 17.5 \text{ miles per hour}
\][/tex]
- Substitute [tex]\( b = 17.5 \text{ mph} \)[/tex] back into the first equation to find [tex]\( c \)[/tex]:
[tex]\[
17.5 + c = 25
\][/tex]
[tex]\[
c = 25 - 17.5 = 7.5 \text{ miles per hour}
\][/tex]
6. Conclude:
- The speed of the boat in still water [tex]\( (b) \)[/tex] is [tex]\( 17.5 \text{ mph} \)[/tex].
- The speed of the current [tex]\( (c) \)[/tex] is [tex]\( 7.5 \text{ mph} \)[/tex].
Thus, the speed of the boat in still water is 17.5 miles per hour.
