help me find the distance between cities?

Question

50mph traveling from Denver to Minneapolis, returning by a different route that covered the same number of miles, average was 55 mph what is the distance between two cities if the total traveling time was 32 hours?

Answer 1

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so , distance = rate*time

so, setting up two equations. One for the traveling from Denver to Minneapolis, and one for the traveling from Minneapolis to Denver, you get:

D1 = R1 * T1

D2 = R2 * T2

but you're told that D1=D2 so, just use D instead of D1 and D2 to get:

D = R1 * T1

D = R2 * T2

the rates are given, so you know R1 = 50 and R2 = 55 so you get:

D = 50T1

D = 55T2

and you're also told that T1+T2 = 32 so you now have three equations in three unknowns:

D = 50T1

D = 55T2

T1+T2 = 32

so rewrite as:

50T1 + 0T2 - D = 0

0T1 + 55T2 - D = 0

1T1 + 1T2 + 0D = 32

set up a matrix:

[50 0 -1 | 0]

[0 55 -1 | 0]

[1 1 0 | 32]

swtich the bottom and top to get

[1 1 0 | 32]

[0 55 -1 | 0]

[50 0 -1 | 0]

now multiply the top by -50 and add those resutls to bottom to get the new bottom matrix of:

[1 1 0 | 32]

[0 55 -1 | 0]

[0 -50 -1 | -1600]

multiply the bottom by -1 to get rid of negative signs

[1 1 0 | 32]

[0 55 -1 | 0]

[0 50 1 | 1600]

now divide the middle by 55 to get

[1 1 0 | 32]

[0 1 -1/55 | 0]

[0 50 1 | 1600]

now multiply the middle by -1 and add to top to get

[1 0 1/55 | 32]

[0 1 -1/55 | 0]

[0 50 1 | 1600]

now multiply the middle by -50 and add to bottom:

[1 0 1/55 | 32]

[0 1 -1/55 | 0]

[0 0 21/11 | 1600]

now multiply bottom by 11/21 to get

[1 0 1/55 | 32]

[0 1 -1/55 | 0]

[0 0 1 | 800]

so the bottom line now shows you that the distance is 800 miles.

Answer 2

distance = speed x time for each leg.

d = 55t1 and d = 50t2

t1 + t2 = 32

So d/55 + d/50 = 32

105d = 32 x 50 x 55 = 88000

d = 838 miles, approx

Answer 3

50+55/2*32/2=840miles distance from cities...i think

Answer 4

distance = rate * time

2d = 55mph * 32 = 1760 miles

distance = 1760/2 = 880 miles

Answer 5

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