Joe and Jodi go canoeing. They paddle upstream 12 miles. On the return strip (with the current) they average 3MPH faster. The total trip takes 8 hours.
1. Find their speed going upstream
2. Find the amount of time it took them to rerun to their starting point
time = distance/speed
So, if their upstream rowing speed is s, then
12/s + 12/(s+3) = 8
s = 3/√2 = 2.12 mi/hr
12/(3/√2 + 3) = 2.34 hours
To find the speed going upstream, we can use the formula: distance = speed * time.
Let's assume the speed of Joe and Jodi going upstream is 'x' miles per hour. The distance they paddle upstream is given as 12 miles. So, the equation becomes:
12 = x * t1
Similarly, on the return trip, they are going downstream with the current, so their speed would be 'x + 3' miles per hour. The distance covered is still 12 miles. So, the equation for the return trip becomes:
12 = (x + 3) * t2
We also know that the total trip takes 8 hours, which means the time taken for the upstream and downstream trips combined should be 8 hours:
t1 + t2 = 8
Now, we can solve these equations simultaneously to find the values of 'x' and 't2'.
1. Solve equation 1 for 't1':
t1 = 12 / x
2. Substitute t1 in equation 3:
12 / x + t2 = 8
3. Solve equation 2 for 't2':
t2 = 8 - 12 / x
4. Substitute t2 in equation 2 and simplify:
12 = (x + 3) * (8 - 12 / x)
To solve this equation, multiply through by 'x' to eliminate the denominator:
12x = (x + 3) * (8x - 12)
Expand the right side:
12x = 8x^2 - 12x + 24x - 36
Combine like terms:
0 = 8x^2 + 12x - 12x - 24x - 36
Simplify further:
0 = 8x^2 - 24x - 36
Now, you can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Since factoring might not work, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Using the quadratic formula, we get:
x = (-(-24) ± √((-24)^2 - 4 * 8 * (-36))) / (2 * 8)
Simplify:
x = (24 ± √(576 + 1152)) / 16
x = (24 ± √(1728)) / 16
x = (24 ± 24√3) / 16
Simplify further:
x = (6 ± 6√3) / 4
x = (3 ± 3√3) / 2
Thus, the possible speeds going upstream are (3 + 3√3) / 2 mph and (3 - 3√3) / 2 mph.
To find the amount of time it took them to return to their starting point, we can substitute the value of 'x' into t2 = 8 - 12 / x, using one of the two possible values for 'x' obtained above.