Create two linear models for Janelle and Swetha's questions answered since Janelle started. Graph the two equations, using your calculator as needed, and solve the problem.
Understand the Problem
The question is asking to create linear equations representing the number of questions answered by Janelle and Swetha, and to determine when Janelle catches up with Swetha based on their rates of answering questions. It also requires graphing these equations and solving the problem.
Answer
Janelle catches up with Swetha after 16 minutes, when both have answered 32 questions.
Answer for screen readers
Janelle catches up with Swetha after 16 minutes, when both have answered 32 questions.
Steps to Solve
- Create Janelle's Equation
Janelle answers questions at a rate of 2 questions per minute. Since she starts at 0 questions answered, her equation is:
$$ y = 2x $$
where ( y ) represents the number of questions answered and ( x ) represents the minutes since she started.
- Create Swetha's Equation
Swetha had already completed 12 questions by the time Janelle started. She answers at a rate of 5 questions every 4 minutes, which is ( \frac{5}{4} = 1.25 ) questions per minute. Her equation is:
$$ y = \frac{5}{4}x + 12 $$
- Set the two equations equal to find when Janelle catches up
To find when Janelle catches up with Swetha, set the equations equal to each other:
$$ 2x = \frac{5}{4}x + 12 $$
- Solve for ( x )
First, isolate ( x ) on one side:
Multiply through by 4 to eliminate the fraction:
$$ 4(2x) = 5x + 48 $$
This simplifies to:
$$ 8x = 5x + 48 $$
Now, subtract ( 5x ) from both sides:
$$ 3x = 48 $$
Next, divide by 3:
$$ x = 16 $$
- Find the number of questions answered at that time
Substitute ( x = 16 ) into Janelle's equation:
$$ y = 2(16) = 32 $$
So, when Janelle catches up with Swetha, she has answered 32 questions.
Janelle catches up with Swetha after 16 minutes, when both have answered 32 questions.
More Information
At the 16-minute mark, Janelle has answered the same number of questions as Swetha. Janelle's faster rate allows her to eventually catch up, starting from a position of zero questions, while Swetha had a head start of 12 questions.
Tips
- Incorrect Rate Calculation: Be careful with converting rates; ensure understanding of how to express the rate per minute.
- Not Setting Equations Equal: Missing the step to set the two equations equal to find the intersection point.
- Algebra Errors: Watch for errors when moving terms from one side of the equation to the other.
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