Hector is in a 42-mile bicycle race, having completed 18 miles at a speed of 12 miles per hour when Wanda starts at 16 miles per hour. 1) Write and solve an equation to find when W... Hector is in a 42-mile bicycle race, having completed 18 miles at a speed of 12 miles per hour when Wanda starts at 16 miles per hour. 1) Write and solve an equation to find when Wanda will catch up to Hector. 2) Will Wanda catch up before the race is complete? Explain. 3) What constant speed could Wanda travel to catch up with Hector at the finish line? Explain.
Understand the Problem
The question presents a scenario involving two cyclists, Hector and Wanda, and asks for a mathematical analysis of when Wanda will catch up to Hector, whether she will catch him before he finishes the race, and what speed she would need to catch him at the finish line. It requires setting up equations based on their speeds and distances covered.
Answer
Wanda will catch up to Hector in \( 4.5 \) hours.
Answer for screen readers
Wanda will catch up to Hector in ( t = 4.5 ) hours.
Steps to Solve
-
Define Variables Let ( t ) represent the time (in hours) it takes Wanda to catch up to Hector.
-
Write Hector's Distance Equation Hector has already completed 18 miles and is traveling at 12 miles per hour. Thus, the distance he covers in time ( t ) is: $$ D_H = 18 + 12t $$
-
Write Wanda's Distance Equation Wanda starts from 0 miles and travels at 16 miles per hour. Thus, her distance covered in time ( t ) is: $$ D_W = 16t $$
-
Set Distances Equal To find when Wanda catches Hector, set their distance equations equal: $$ 18 + 12t = 16t $$
-
Solve for ( t ) Rearranging the equation gives: $$ 18 = 16t - 12t $$ $$ 18 = 4t $$ Dividing both sides by 4: $$ t = \frac{18}{4} = 4.5 $$
-
Interpret the Result This means Wanda will catch up to Hector 4.5 hours after she starts.
Wanda will catch up to Hector in ( t = 4.5 ) hours.
More Information
Wanda starts her ride after Hector has already covered some distance. The calculations show that she can catch up to him due to her higher speed, which ultimately helps solve the question regarding the scenario.
Tips
- Ignoring Starting Distances: Make sure to account for the distance Hector has already traveled.
- Incorrectly Rearranging Equations: Be careful with algebraic manipulations, especially when isolating the variable.
AI-generated content may contain errors. Please verify critical information
