Laura is training for a triathlon, but she doesn't feel like swimming. She bikes 20 miles at 2x+1 miles per hour, spends five minutes parking her bike, changing to her running shoes, and catching a drink of water. Then, she runs 5 miles at x miles per hour. Her total workout lasts 110 minutes. How fast did Laura run, to the nearest hundredth of a mile per hour?
Solve for x:
1/12 + 5/x + 20/(2 x + 1) = 11/6
Bring 1/12 + 5/x + 20/(2 x + 1) together using the common denominator 12 x (2 x + 1):
(2 x^2 + 361 x + 60)/(12 x (2 x + 1)) = 11/6
Cross multiply:
6 (2 x^2 + 361 x + 60) = 132 x (2 x + 1)
Expand out terms of the left hand side:
12 x^2 + 2166 x + 360 = 132 x (2 x + 1)
Expand out terms of the right hand side:
12 x^2 + 2166 x + 360 = 264 x^2 + 132 x
Subtract 264 x^2 + 132 x from both sides:
-252 x^2 + 2034 x + 360 = 0
Divide both sides by -252:
x^2 - (113 x)/14 - 10/7 = 0
Add 10/7 to both sides:
x^2 - (113 x)/14 = 10/7
Add 12769/784 to both sides:
x^2 - (113 x)/14 + 12769/784 = 13889/784
Write the left hand side as a square:
(x - 113/28)^2 = 13889/784
Take the square root of both sides:
x - 113/28 = sqrt(13889)/28 or x - 113/28 = -sqrt(13889)/28
Add 113/28 to both sides:
x = 113/28 + sqrt(13889)/28 or x - 113/28 = -sqrt(13889)/28
Add 113/28 to both sides:
x = 113/28 + sqrt(13889)/28 =8.24 - miles per hour.
+36916 rate * time = distance
distance/rate = time
105 minutes moving time Bike time = distance/rate = 20/(2x+1)
Run time = 5/x
added together the time = 105 min = 1.75 hr
20/(2x+1) + 5/x = 1.75
20x /(2x+1) + 5 = 1.75 x
20 x + 5(2x+1) = 1.75 x (2x+1)
20 x + 10x + 5 = 3.5 x2 + 1.75x
3.5 x^2 -28.25x -5 =0
x = 8.24 m/hr running speed (via quadratic formula)
