Algebra Word Problem

Question

The town of Oldberg is 20 miles east and 15 miles north of Reggie's home. Reggie goes on a bike ride and travels 8 miles east and 15 miles south every hour. He starts in the morning and passes through Oldberg at noon. After traveling for a while, he is 39 miles from home.

A: How long after Reggie passes through Oldberg is he 39 miles from home?

B: Reggie was also 39 miles from home earlier in the morning before he passed through Oldberg. To the nearest minute, when was this?

Thank you so much!

Answer 1

The given region can be formulated as a linear programming problem as follows:

Maximize z = 0x + 0y + 0z + 0s1 + 0s2 + 0s3

Subject to:

x - s1 = 1 -2x + y + s2 = 5 x + y - z + s3 = 2 x, y, z, s1, s2, s3 ≥ 0

We can see that the problem is already in the form of slack variables. The basic solution obtained by choosing the slack variables as pivots is:

x = 1 y = 3 z = 0 s1 = 0 s2 = 0 s3 = 1

Therefore, the basic solution is (1,3,0,0,0,1).

Answer 2

If Oldberg is north (and east) of Reggie's home and Reggie is traveling south (and east), he is not going to go through Oldberg, at least until he goes all the way around the world! Check and make sure the problem statement is correct. Also, his path does not go directly through Oldberg, presuming he were headed north instead of south.

Reggie's diagonal speed is:

√(8² + 15²) = √(64+225) = √(289) =17 miles per hour

distance = rate × time

39 miles = (17 mph)×time

time = (39 miles)/(17 mph) = 2 5/17 hours, which is approximately 2 hours, 17 minutes, and 39 seconds.

If Reggie was headed straight to Oldberg (which he is not), he would be 25 miles away:

√(15² + 20²) = √(225+400) = √(625) = 25

39 - 25 = 14 miles past Oldberg.

14/17 = 49 minutes and 25 seconds, approximately.

If Reggie was 39 miles away fom Oldberg, it would be almost 2 hours, 17 minutes, and 39 seconds earlier. That would be 9:42:21 am, or 9:42 am to the nearest minute.