Write and solve an equation to find when Wanda will catch up to Hector.
Understand the Problem
The question asks us to form and solve an equation to determine when Wanda will catch up to Hector during a bicycle race, given their speeds and the distance both have traveled.
Answer
Wanda will catch up to Hector in \( 4.5 \) hours after her start.
Answer for screen readers
Wanda will catch up to Hector in ( 4.5 ) hours after she starts, which is ( 6 ) hours after Hector began the race.
Steps to Solve
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Identify the distances traveled by Hector and Wanda
To find when Wanda catches up to Hector, we first need to express the distances they have traveled as functions of time.
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Let ( t ) be the time in hours that Wanda has been riding.
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Hector has a 18-mile head start, so the distance he has traveled when Wanda has been riding for ( t ) hours is: $$ D_H = 18 + 12t $$ where ( 12 ) is Hector's speed.
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The distance Wanda travels in ( t ) hours is: $$ D_W = 16t $$ where ( 16 ) is Wanda's speed.
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Set the distances equal to each other
We want to find the time when Wanda catches up to Hector, so we set the two distance equations equal to each other: $$ 18 + 12t = 16t $$
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Solve for ( t )
Rearrange the equation to isolate ( t ):
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First, subtract ( 12t ) from both sides: $$ 18 = 16t - 12t $$ $$ 18 = 4t $$
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Now, divide both sides by 4: $$ t = \frac{18}{4} $$ $$ t = 4.5 $$
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Conclusion regarding the distance and time
To find the total time from the start of the race until Wanda catches up to Hector, we should add the time Hector has already been riding (which is ( 1.5 ) hours, calculated as ( 18 \text{ miles} \div 12 \text{ mph} = 1.5 \text{ hours} )): $$ \text{Total time} = t + 1.5 = 4.5 + 1.5 = 6 \text{ hours} $$
Wanda will catch up to Hector in ( 4.5 ) hours after she starts, which is ( 6 ) hours after Hector began the race.
More Information
This problem demonstrates principles of relative motion and how to set up equations based on distances traveled over time. The simplicity of the speeds allows us to see how differences in speed affect when one person will catch up to another.
Tips
- Not accounting for the head start: Remember to consider the distance that Hector has already traveled when setting up the equation.
- Incorrect algebra when isolating ( t ): Pay close attention to operations when rearranging equations.
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