Mikayla is training for a half-marathon. At the beginning of her training, she is able to run 2 miles without stopping. Her goal is to increase her distance by three quarters of a mile each week.
a) Write an equation to represent the number of miles Mikayla is able to run after each week. Define your variables.
b) If a half-marathon (13.1 miles) is 12 weeks after the start of Mikayla's training program, will she be ready?The Correct Answer and Explanation is:
Let’s solve this in two parts:
a) Write an equation to represent the number of miles Mikayla is able to run after each week. Define your variables.
Let:
- ww = number of weeks since Mikayla started training
- M(w)M(w) = number of miles Mikayla can run after ww weeks
She starts at 2 miles and increases her distance by three-quarters of a mile (0.75 miles) each week. This is an arithmetic sequence, and the equation to model this is: M(w)=2+0.75wM(w) = 2 + 0.75w
b) If a half-marathon (13.1 miles) is 12 weeks after the start of Mikayla’s training program, will she be ready?
Substitute w=12w = 12 into the equation: M(12)=2+0.75(12)=2+9=11 milesM(12) = 2 + 0.75(12) = 2 + 9 = 11 \text{ miles}
So, after 12 weeks, Mikayla will be able to run 11 miles.
A half-marathon is 13.1 miles, which means: 13.1−11=2.1 miles short13.1 – 11 = 2.1 \text{ miles short}
Explanation (300+ words):
To determine Mikayla’s readiness for a half-marathon, we start by modeling her weekly progress. Mikayla begins by running 2 miles and adds three-quarters of a mile every week. This constant weekly increase indicates that her training follows an arithmetic pattern. The general form of an arithmetic sequence is: M(w)=initial value+(rate of change)×wM(w) = \text{initial value} + (\text{rate of change}) \times w
For Mikayla, the initial value is 2 miles, and the rate of change is 0.75 miles per week. So, her running capability each week is modeled by: M(w)=2+0.75wM(w) = 2 + 0.75w
Now, the question is whether she can run 13.1 miles (the distance of a half-marathon) by week 12. Plugging in w=12w = 12 into the equation: M(12)=2+0.75(12)=2+9=11 milesM(12) = 2 + 0.75(12) = 2 + 9 = 11 \text{ miles}
This means that after 12 weeks of training, Mikayla will be able to run 11 miles continuously. However, since a half-marathon is 13.1 miles, she falls short by 2.1 miles. While 11 miles is a strong achievement, it’s not quite the full race distance.
In conclusion, Mikayla will not be fully ready to run a half-marathon after 12 weeks at her current rate of increase. To be fully prepared, she would need more time—specifically, we can solve: 13.1=2+0.75w⇒0.75w=11.1⇒w=11.10.75=14.8 weeks13.1 = 2 + 0.75w \Rightarrow 0.75w = 11.1 \Rightarrow w = \frac{11.1}{0.75} = 14.8 \text{ weeks}
So she would need about 15 weeks total to be ready.