olve the following problem and show all work: Patrick can run 3 miles in 30 minutes, and Paul can run 4 miles in 32 minutes. Find the constants of proportionality for Patrick and Paul, and determine who runs faster.
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in each case, k = distance/time
To find the constants of proportionality for Patrick and Paul, we need to determine their rates of running.
Rate of running can be calculated by dividing the distance covered by the time taken. So, to find the rate of running for Patrick, we divide the distance he runs (3 miles) by the time taken (30 minutes).
Rate of running for Patrick = Distance / Time = 3 miles / 30 minutes = 0.1 miles per minute.
Similarly, to find the rate of running for Paul, we divide the distance he runs (4 miles) by the time taken (32 minutes).
Rate of running for Paul = Distance / Time = 4 miles / 32 minutes = 0.125 miles per minute.
Now, the constants of proportionality for Patrick and Paul are equal to their respective rates of running.
Constant of proportionality for Patrick = 0.1 miles per minute.
Constant of proportionality for Paul = 0.125 miles per minute.
To determine who runs faster, we compare their rates of running. Since Paul has a higher rate of running (0.125 miles per minute) compared to Patrick (0.1 miles per minute), Paul runs faster than Patrick.