Jean runs 6 mi and then rides 24 mi on her bicycle in a biathlon. She rides 8 mph faster than she runs. determine both average speeds if total time is 2.25 hr

Question

college algebra, solving rational equations.  Cant seem to get the formula written correctly.

Answer 1

Speed is the distance over time.

 

Let x = the time riding bike

Let y = the time running

 

We know that the difference between the bike speed and running speed is 8 mph. We also know that the total time is 2.25 hr.  Knowing these facts, we can set equations.

 

 

x + y = 2.25                   eq1

 

(24 / x) - (6 / y) = 8          eq2

 

 

After substituting eq1 into eq2, we get

 

(24 / x) - (6 / (2.25 - x)) = 8

 

 

We have a rational equation with different denominators, so we need to use the LCD.  LCD is   x(2.25 - x).

 

 

[24(2.25 - x) - 6x] / (x(2.25 - x)) = [8x(2.25 - x)] / (x(2.25 - x))

 

 

Equate numerators to solve for x.

 

24(2.25 - x) - 6x = 8x(2.25 - x)

 

54 - 30x = 18x - 8x²

 

 

Add 8x² and subtract 18x on both sides of the equation.

 

8x² - 48x + 54 = 0

 

 

We now have a quadratic equation.   Factor out a 2.

 

2(4x² - 24x + 27) = 0

 

 

Set the term in parenthesis equal to zero.

 

 

4x² - 24x + 27 = 0

 

 

Use the quadratic formula to solve for x:

 

x = (-b ± √(b² - 4ac)) / 2a

 

where:

a = 4

b = -24

c = 27

 

 

Plug in these values into the formula.  You will have two solutions because of the plus/minus sign.  Keep in mind that both

 

x ≤ 0

x = 2.25

 

cannot be solutions, since zero cannot be in the denominator of the original equation, and time is always positive.

 

 

Once you have your x value, plug it into

 

24 / x          and          6 / (2.25 - x)

 

 

to get the running speed and bike speed.