Question

Denise has 6 hours to spend training for an upcoming race. She completes her training by running full speed the distance of the race and walking back the same distance to cool down. If she runs at a speed of 9mph and walks back at a speed of 3mph, how long should she plan to spend walking back?

Answer 1

To solve this problem, we need to recall the formula d = rd (distance = rates x time)

She runs at a speed of 7 mph and walks at a speed of 3 mph.

Her distance running is

d = 7tr

where tr is the time she spends running

Her distance walking is

d = 3tw

where tw is the time she spends walking

The distances are the same so

7tr = 3tw

We also know that the total time is 4 hours

tr + tw = 4

tr = 4-tw

Substitute this value of tr in the first equation

7tr = 3tw

7(4-tw) = 3tw

28-7tw = 3tw

28 = 10tw

2.8 = tw

Denise will spend 2.8 hours (2 hours, 48 minutes) walking back and 1.2 hours (1 hour, 12 minutes running.

Answer 2

Explanation:

Let's call the distance of the race "d".

When Denise runs the distance of the race at a speed of 9 mph, she takes:

time = distance / speed = d / 9

Then, when she walks back the same distance at a speed of 3 mph, she takes:

time = distance / speed = d / 3

Since she has a total of 6 hours for training, the sum of these two times must be less than or equal to 6 hours:

d/9 + d/3 ≤ 6

We can simplify this inequality by multiplying both sides by the least common multiple of the denominators, which is 9:

d + 3d ≤ 54

4d ≤ 54

d ≤ 13.5

Therefore, the distance of the race must be at most 13.5 miles.

To find the time Denise should plan to spend walking back, we can use the time formula for walking back:

time = distance / speed = d / 3

Since Denise has already spent time running the distance of the race, her total training time will be:

total time = time running + time walking back

total time = d/9 + d/3

We want to find the time for walking back, so we can solve this equation for d/3:

d/3 = total time - d/9

d/3 = (9total time - d)/9

4d/9 = 9total time/9

d = 36total time/13

Now we can substitute this expression for d into the time formula for walking back:

time walking back = d/3 = (36total time/13) / 3 = 12total time/13

Therefore, Denise should plan to spend 12/13 of her total training time walking back. For example, if she plans to train for 6 hours, she should spend about 5.54 hours (or 5 hours and 32 minutes) walking back.