gregor is planning a driving trip where he knows that half his trip will be along major highways and part on country roads. The speed limit is 100km/h on highways and 60km/h on other roads. the total time he wants to spend driving is 12h. I set up the equation: 10h+6r=12, but I don't know how to solve for 3 more combinations of times gregor could spend driving on highways and country roads. please help
He is going some distance, call it 2d
then
he goes 1d at 100 km/hr
and
he goes 1 d at 60 km/hr
time = distance / speed
so
d/100 + d/60 = 12
multiply both sides by 300
3 d + 5 d = 3600
8 d = 3600
d = 450
so
2 d = 900 km, the distance he goes
To solve for the different combinations of times Gregor could spend driving on highways and country roads, we can use the equation you have already set up: 10h + 6r = 12.
Let's rearrange the equation to solve for h (time spent on highways):
10h + 6r = 12
Subtract 6r from both sides:
10h = 12 - 6r
Divide both sides by 10:
h = (12 - 6r)/10
Now we can substitute different values for r (time spent on country roads) and calculate the corresponding value for h:
1. For r = 1:
h = (12 - 6(1))/10
h = (12 - 6)/10
h = 6/10
h = 0.6 hours
Therefore, Gregor could spend 0.6 hours (or 36 minutes) driving on highways and 1 hour driving on country roads.
2. For r = 2:
h = (12 - 6(2))/10
h = (12 - 12)/10
h = 0 hours
In this case, Gregor would not spend any time on highways, as he would be driving only on country roads.
3. For r = 3:
h = (12 - 6(3))/10
h = (12 - 18)/10
h = -6/10
h = -0.6 hours
A negative value means it is not a valid solution since time cannot be negative. Thus, there is no valid combination of 3 hours driving on country roads and 3 hours driving on highways in a total of 12 hours.
So, the three different combinations of times Gregor could spend driving on highways and country roads are:
- 0.6 hours on highways and 1 hour on country roads
- 0 hours on highways and 2 hours on country roads
- No valid combination for 3 hours on highways and 3 hours on country roads.