Elena bikes 20 minutes each day for exercise. Write an equation to describe the relationship between her distance in miles, D, and her biking speed, in miles per hour, when she bikes: A.) At a constant speed of 13 miles per hour for the entire 20 minutes. B.) At a constant speed of 15 miles per hour for the first 5 minutes, then at 12 miles per hour for the last 15 minutes. C.) At a constant speed of M miles per hour for the first 5 minutes, then N miles per hour for the last 15 minutes. Plz answer ASAP!
The Correct Answer and Explanation is:
Here are the correct equations and explanations for each part of the problem:
A. Constant Speed of 13 mph for 20 Minutes
We use the formula: Distance=Speed×Time\text{Distance} = \text{Speed} \times \text{Time}
Time must be in hours. Since 20 minutes = 20/60 = 1/3 hour, the equation is: D=13×13=133≈4.33 milesD = 13 \times \frac{1}{3} = \frac{13}{3} \approx 4.33 \text{ miles}
B. 15 mph for 5 minutes, then 12 mph for 15 minutes
Convert time to hours:
- 5 minutes = 5/60 = 1/12 hour
- 15 minutes = 15/60 = 1/4 hour
Use the formula for each segment, then add distances: D=(15×112)+(12×14)=1512+124=54+3=174=4.25 milesD = (15 \times \frac{1}{12}) + (12 \times \frac{1}{4}) = \frac{15}{12} + \frac{12}{4} = \frac{5}{4} + 3 = \frac{17}{4} = 4.25 \text{ miles}
C. M mph for 5 minutes, then N mph for 15 minutes
Using the same time conversions as part B: D=M×112+N×14=M12+N4D = M \times \frac{1}{12} + N \times \frac{1}{4} = \frac{M}{12} + \frac{N}{4}
Explanation
To solve these types of problems, we rely on the fundamental formula for distance: Distance=Speed×Time\text{Distance} = \text{Speed} \times \text{Time}
This equation means that the total distance Elena bikes depends on how fast she goes (speed) and how long she rides (time). However, because speed is measured in miles per hour (mph), time must also be expressed in hours rather than minutes. That’s why in all parts of the problem, we first convert minutes into hours. For example, 20 minutes is 13\frac{1}{3} of an hour, and 5 minutes is 112\frac{1}{12} of an hour.
In part A, since Elena bikes at a constant speed the whole time, we directly multiply the speed by the time to get the distance.
In part B, she changes her speed, so we break the trip into two parts. We calculate how far she bikes at each speed and then add the two distances together.
In part C, we don’t use specific numbers for the speeds but use variables MM and NN. The process is the same: convert time, calculate each segment, and add them.
Understanding this method helps in real-life scenarios too—when tracking workouts, planning trips, or analyzing pace. The key is consistent units and applying the basic distance formula logically.
