Elizabeth visits her friend Andrew and then returns home by the same route. She always walks 2km/h when going uphill, 6km/h when going downhill and 3km/h when on level ground. If her total walking time is 6 hours, then what is the total distance she walks in km?
She walks 18km in total.
using u,d,f for uphill, downhill, flat distances on the outward journey,
recall that time = distance/speed
she walks 9 km in each direction
(u/2 + f/3 + d/6) + (d/2 + f/3 + u/6) = 6
2u/3 + 2f/3 + 2d/3 = 6
u + f + d = 9
you are correct. the round trip is 18 km.
To find the total distance Elizabeth walks, we need to determine the time she spends walking uphill, downhill, and on level ground.
Let's assume Elizabeth walks uphill for x hours, downhill for y hours, and on level ground for z hours.
We know that Elizabeth walks at a speed of 2 km/h uphill, 6 km/h downhill, and 3 km/h on level ground. We can use the formula Distance = Speed x Time to find the distance she walks in each scenario.
Distance uphill = 2 km/h * x hours = 2x km
Distance downhill = 6 km/h * y hours = 6y km
Distance on level ground = 3 km/h * z hours = 3z km
We are given that the total walking time is 6 hours, so we can write the equation: x + y + z = 6
We can also write the equation for the total distance walked: Distance uphill + Distance downhill + Distance on level ground = Total distance walked
2x + 6y + 3z = Total distance walked
Since we want to find the total distance, we can substitute the total distance given in the question: 2x + 6y + 3z = 18 km
Now we have a system of two equations:
x + y + z = 6
2x + 6y + 3z = 18
We can solve this system of equations by substitution or elimination. Let's use the elimination method to solve for z.
Multiplying the first equation by 3, we get:
3x + 3y + 3z = 18
Subtracting this equation from the second equation, we eliminate z:
(2x + 6y + 3z) - (3x + 3y + 3z) = 18 - 18
-1x + 3y = 0
or
x = 3y
Substituting x = 3y into the first equation:
(3y) + y + z = 6
4y + z = 6
z = 6 - 4y
Now, we can express the total distance in terms of y:
2x + 6y + 3z = 18
2(3y) + 6y + 3(6 - 4y) = 18
6y + 6y + 18 - 12y = 18
12y - 12y = 18 - 18
0 = 0
This means that y can be any value. Let's choose y = 1 for simplicity.
Substituting y = 1 into z = 6 - 4y:
z = 6 - 4(1)
z = 6 - 4
z = 2
Now, we can find x using x = 3y:
x = 3(1)
x = 3
So, Elizabeth walks uphill for 3 hours, downhill for 1 hour, and on level ground for 2 hours.
Finally, we can calculate the distance:
Distance uphill = 2x km = 2(3) = 6 km
Distance downhill = 6y km = 6(1) = 6 km
Distance on level ground = 3z km = 3(2) = 6 km
Therefore, the total distance Elizabeth walks is 6 km + 6 km + 6 km = 18 km.