Certainly, let's carefully analyze the given situation and equations. Sun is paddling upstream and downstream with the following details:
1. Distance to paddle upstream: 8 miles
2. Time taken to paddle upstream: 2 hours
3. Distance to paddle downstream: 8 miles
4. Time taken to paddle downstream: 1 hour
We are given two equations involving the paddling speed [tex]\( x \)[/tex] and the current speed [tex]\( y \)[/tex]:
[tex]\[
2(x - y) = a
\][/tex]
[tex]\[
b(x + y) = 8
\][/tex]
First, let's determine the speeds:
### Determining the Speeds
The effective speed going upstream (against the current) is:
[tex]\[ x - y = \frac{8 \text{ miles}}{2 \text{ hours}} = 4 \text{ mph} \][/tex]
The effective speed going downstream (with the current) is:
[tex]\[ x + y = \frac{8 \text{ miles}}{1 \text{ hour}} = 8 \text{ mph} \][/tex]
### Using the Equations
Now, let's substitute these effective speeds into the given equations to find [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
#### Equation 1
[tex]\[ 2(x - y) = a \][/tex]
Since [tex]\( x - y = 4 \)[/tex],
[tex]\[ 2(4) = a \][/tex]
[tex]\[ a = 8 \][/tex]
#### Equation 2
[tex]\[ b(x + y) = 8 \][/tex]
Since [tex]\( x + y = 8 \)[/tex],
[tex]\[ b(8) = 8 \][/tex]
[tex]\[ b = 1 \][/tex]
### Verifying the True Statements
We now verify which of the given statements are correct:
1. [tex]\( a = 8 \)[/tex] → True
2. [tex]\( b = 8 \)[/tex] → False
3. [tex]\( a = 1 \)[/tex] → False
4. [tex]\( b = 1 \)[/tex] → True
5. [tex]\( a = b \)[/tex] → False (since [tex]\( a = 8 \)[/tex] and [tex]\( b = 1 \)[/tex])
Therefore, the correct options are:
- [tex]\( a = 8 \)[/tex]
- [tex]\( b = 1 \)[/tex]
Thus, the true statements are:
[tex]\[ a = 8 \][/tex]
[tex]\[ b = 1 \][/tex]
