SOLUTION: Liz commutes 30 mi. to her job each day. She Finds that if she drives 10 mi/h faster it takes her 6 minutes less to get to work. Find her new speed. I tried setting this problem

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Question 370575: Liz commutes 30 mi. to her job each day. She Finds that if she drives 10 mi/h faster it takes her 6 minutes less to get to work. Find her new speed.
I tried setting this problem up on a chart. For some reason when I worked the problem out I came out with really odd fractions, and I am not quite sure what I have done wrong. Any kind of help would truly mean a lot to me.
Found 2 solutions by ewatrrr, jim_thompson5910:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!


Hi,

Let r represent her old speed

d = r*t  

30 = r*t

30/r = t

30%2F%28r+%2B+10mph%29+=+t+-+1%2F10

30%2F%28r+%2B+10mph%29+=+%2830%2Fr%29+-+%281%2F10%29

Multiplying thru by 10r(r+10) so as all denominators = 1

300r = 300(r+10) - r(r+10)

300r = 300r + 3000 - r^ - 10r

r^2 + 10r - 3000 = 0

factor

(x + 60)(x-50) = 0

x + 60)= 0  x = -60 cannot use

(x-50) = 0  x = 50mph her old speed.  Her new speed would be 60mph( 10mph faster)

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let r = original speed.

and let t = time to get to work driving that original speed.


So if "Liz commutes 30 mi. to her job each day", this means that 30=rt because we're using the formula d=rt where the distance is d=30


So the first equation is 30=rt


In addition, because "she drives 10 mi/h faster it takes her 6 minutes less to get to work", we can say that 30=%28r%2B10%29%28t-1%2F10%29

Notes: since she drives 10 mph faster, her new speed is r+10. Also, because 60 min = 1 hour, this means that 6 min = 6%2F60=1%2F10 hours. So if it takes 6 mins or 1%2F10 or an hour less, then the new time is t-1%2F10


So the second equation is 30=%28r%2B10%29%28t-1%2F10%29


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Now let's use both equations to solve for t and r


30=rt Start with the first equation.


30%2Ft=r Divide both sides by t to isolate r.


r=30%2Ft Flip the equation.


30=%28r%2B10%29%28t-1%2F10%29 Move onto the second equation.


30=%2830%2Ft%2B10%29%28t-1%2F10%29 Plug in r=30%2Ft


30=%2830%2Ft%2B10t%2Ft%29%28t-1%2F10%29 Multiply 10 by t%2Ft


30=%28%2830%2B10t%29%2Ft%29%28t-1%2F10%29 Combine the fractions.


30=%28%2830%2B10t%29%2Ft%29%28t%29-%28%2830%2B10t%29%2Ft%29%281%2F10%29 Distribute


30=30%2B10t-%283%2Bt%29%2Ft Multiply.


Note: the 't' terms cancel in the first fraction while in the second, we're dividing each term by 10.


30t=30t%2B10t%5E2-%283%2Bt%29 Multiply EVERY term by the LCD 't' to clear out the fractions.


30t=30t%2B10t%5E2-3-t Distribute.


0=30t%2B10t%5E2-3-t-30t Subtract 30t from both sides.


0=10t%5E2-t-3 Combine like terms.



Notice that the quadratic 10t%5E2-t-3 is in the form of At%5E2%2BBt%2BC where A=10, B=-1, and C=-3


Let's use the quadratic formula to solve for "t":


t+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


t+=+%28-%28-1%29+%2B-+sqrt%28+%28-1%29%5E2-4%2810%29%28-3%29+%29%29%2F%282%2810%29%29 Plug in A=10, B=-1, and C=-3


t+=+%281+%2B-+sqrt%28+%28-1%29%5E2-4%2810%29%28-3%29+%29%29%2F%282%2810%29%29 Negate -1 to get 1.


t+=+%281+%2B-+sqrt%28+1-4%2810%29%28-3%29+%29%29%2F%282%2810%29%29 Square -1 to get 1.


t+=+%281+%2B-+sqrt%28+1--120+%29%29%2F%282%2810%29%29 Multiply 4%2810%29%28-3%29 to get -120


t+=+%281+%2B-+sqrt%28+1%2B120+%29%29%2F%282%2810%29%29 Rewrite sqrt%281--120%29 as sqrt%281%2B120%29


t+=+%281+%2B-+sqrt%28+121+%29%29%2F%282%2810%29%29 Add 1 to 120 to get 121


t+=+%281+%2B-+sqrt%28+121+%29%29%2F%2820%29 Multiply 2 and 10 to get 20.


t+=+%281+%2B-+11%29%2F%2820%29 Take the square root of 121 to get 11.


t+=+%281+%2B+11%29%2F%2820%29 or t+=+%281+-+11%29%2F%2820%29 Break up the expression.


t+=+%2812%29%2F%2820%29 or t+=++%28-10%29%2F%2820%29 Combine like terms.


t+=+3%2F5 or t+=+-1%2F2 Simplify.


So the possible solutions are t+=+3%2F5 or t+=+-1%2F2


However, since a negative time isn't possible, this means that the only solution for 't' is t=3%2F5


Remember that 't' is the time in hours. So convert to minutes to get %283%2F5%29%2A%2860%29+=+180%2F5+=+36 minutes


So the time it takes to travel 30 miles at the original speed is 36 minutes.


Recall that we made r=30%2Ft. So plug t=3%2F5 into the equation to get r=30%2F%283%2F5%29=%2830%2F1%29%285%2F3%29=150%2F3=50


So her original speed is 50 mph. Add 10 mph to this speed to get 50+10 = 60 mph


So her new speed is 60 mph.


I'll leave the check to you. Remember to use the formula d=rt


If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my tutoring website

Jim