Chantal bicycles at a speed of 20 miles per hour and has already ridden 45 miles when Trish begins cycling. Trish bicycles at a speed of 35 miles per hour.
a. Write and solve a system of equations for the given situation.
b. In how many hours will Chantal and Trish have cycled the same amount of miles? c. How many miles will both have cycled?
a) Let t be the time in hours that Trish has been cycling. Then Chantal has been cycling for t + 45/20 hours.
The system of equations can be written as:
Chantal's distance = 20 * (t + 45/20) (since the speed is given in miles per hour)
Trish's distance = 35t
b) To find the time at which both have cycled the same amount of miles, we set the two distances equal to each other:
20 * (t + 45/20) = 35t
c) To find the total distance both have cycled, we can substitute the value of t found in part b) into either of the original equations.
a. Let's assume that both Chantal and Trish start cycling at the same time, t=0.
Let x be the number of hours both Chantal and Trish have cycled.
For Chantal: Distance = Speed * Time
Chantal's distance = 20x + 45
For Trish: Distance = Speed * Time
Trish's distance = 35x
b. To find the number of hours both Chantal and Trish have cycled the same amount of miles, we need to set their distances equal to each other and solve for x.
20x + 45 = 35x
Combine like terms:
45 = 35x - 20x
Simplify:
45 = 15x
Divide both sides by 15:
x = 3
So, they will have cycled the same amount of miles after 3 hours.
c. To find the number of miles both Chantal and Trish will have cycled, substitute the value of x into either equation:
Chantal's distance = 20x + 45
= 20(3) + 45
= 60 + 45
= 105 miles
Therefore, both Chantal and Trish will have cycled 105 miles each.