Question
Kathy can run 6 mi to the beach in the same amount of time Dennis can ride his bike 12 mi to work. Kathy runs 2 mph slower than Dennis rides his bike. Find
their speeds.
Part:
Part 1 of 2
Dennis’ speed is
their speeds.
Part:
Part 1 of 2
Dennis’ speed is
Ask by French Pena. in the United States
Mar 31,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Dennis’ speed is 4 mph.
Solution
To find Dennis’ speed, we can use the information given in the problem. Let’s denote Kathy’s speed as
From the problem, we know that Kathy can run 6 miles to the beach in the same amount of time Dennis can ride his bike 12 miles to work. This means that the time taken by Kathy to run 6 miles is equal to the time taken by Dennis to ride 12 miles.
We also know that Kathy runs 2 mph slower than Dennis rides his bike. This can be represented as
Now, we can use the formula
Let’s denote the time taken by Kathy to run 6 miles as
We have the following equations:
We can solve these equations to find the speeds of Kathy and Dennis.
Solve the system of equations by following steps:
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
- step1: Substitute the value of
- step2: Substitute the value of
- step3: Multiply both sides of the equation by LCD:
- step4: Simplify the equation:
- step5: Swap the sides:
- step6: Move the constant to the right side:
- step7: Subtract the numbers:
- step8: Change the signs:
- step9: Divide both sides:
- step10: Divide the numbers:
- step11: Substitute the value of
- step12: Calculate:
- step13: Substitute the value of
- step14: Calculate:
- step15: Calculate:
- step16: Check the solution:
- step17: Rewrite:
The solution to the system of equations isand .
Therefore, Dennis’ speed is 4 mph.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Mind Expander
Alright, let’s ride this wave of numbers! To solve this, let’s denote Dennis’ speed as
Since speed equals distance over time, we can set up the equation:
Cross-multiplying gives us
Distributing the 12 gives
Now, let’s put on our history hats! The bicycle, invented in the early 19th century, revolutionized transportation and became a significant part of daily life, encouraging leisure activities just like running does today. Both athletes and casual riders enjoy the thrill of movement, making biking and running popular recreational activities across the globe.
And speaking of challenges, common beginner mistakes include neglecting to account for variations in terrain which can impact speeds significantly! Uphill biking or running can be a lot slower than expected. So remember, if you’re gearing up for an outdoor run or ride, always scout your path first for any sneaky slopes!
