Answer/Step-by-step explanation:
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 6 mph. We also know that the total time is 2.75 hr. Knowing these facts, we can set equations.
x + y = 2.75 eq1
(21 / x) - (10 / y) = 6 eq2
After substituting eq1 into eq2, we get
(21 / x) - (10 / (2.75 - x)) = 6
We have a rational equation with different denominators, so we need to use the LCD. LCD is x(2.75 - x).
[21(2.75 - x) - 10x] / (x(2.75 - x)) = [6x(2.75 - x)] / (x(2.75- x))
Equate numerators to solve for x.
21(2.75 - x) - 10x = 6x(2.75 - x)
57.75- 31x = 16.5x-6x^2
Add 6x^2 and subtract 16.5x^2 on both sides of the equation.
6x^2- 16.5x+57.75=0
We now have a quadratic equation. Factor out a 2.
2(3x^2-8.25+28.875)=0
Set the term in parenthesis equal to zero.
(3x^2-8.25+28.875)=0
Use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 3ac)) / 2a
where:
a = 3
b = 8.25
c = 28.875
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 = 2.75
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
21 / x and 8 / (2.75- x)
to get the running speed and bike speed.