Question

two families are taking a vacation together. they each leave at 9 a.m. on saturday morning, driving their cars along the same route to a cabin on a lake.- the jacksons live 40 miles closer to the cabin than the milligans.- the jacksons drive at an average speed of 50 miles per hour, and the milligans drive at an average speed of 60 miles per hour.an equation can be written to represent this situation for each family relating the time, x, they drive and the distance, y, they are from the cabin.decide if each statement is true or false.true falsethe equation for the jacksons is $y = 40x + 50$ and the equation for the chases is $y = x + 60$.the cars will reach the same point in 4 hours.if the cabin is 400 miles away, it will take the jacksons 7 hours and 12 minutes to get there.the two cars will reach the same point at noon.

Explanation:

Step1: Verify family equations

Let $d_J$ = Jacksons' initial distance to cabin, $d_M$ = Milligans' initial distance to cabin. Given $d_J = d_M - 40$. Distance from cabin after $x$ hours: $y = \text{Initial Distance} - \text{Speed} \times x$.
Jacksons: $y_J = d_J - 50x$
Milligans: $y_M = d_M - 60x = (d_J + 40) - 60x$
The given equations $y=40x+50$ and $y=x+60$ do not match this form.

Step2: Check meeting time

Set $y_J = y_M$:
$d_J - 50x = (d_J + 40) - 60x$
$10x = 40 \implies x=4$ hours. They leave at 9 a.m., so 9+4=1 p.m., not 4 hours to same point relative to cabin (they meet at 1 p.m., not that they reach cabin in 4 hours).

Step3: Calculate Jacksons' travel time

Jacksons' initial distance: $d_J = 400$ miles. Time = $\frac{\text{Distance}}{\text{Speed}} = \frac{400}{50} = 8$ hours, not 7h12m.

Step4: Check noon arrival time

Jacksons reach cabin in 8 hours: 9 a.m. +8h=5 p.m. Milligans reach cabin in $\frac{400+40}{60} = \frac{440}{60} \approx7.33$ hours =7h20m, 9a.m.+7h20m=4:20 p.m. Neither reaches at noon, so they can't meet at noon.

Answer:

1. The equation for the Jacksons is $y = 40x + 50$ and the equation for the Milligans is $y = x + 60$: False
2. The cars will reach the same point in 4 hours: False
3. If the cabin is 400 miles away, it will take the Jacksons 7 hours and 12 minutes to get there: False
4. The two cars will reach the same point at noon: False