Caden and his friends set out to sea on their annual fishing trip.

Question

There is a proportional relationship between the time (in hours) Caden and his friends spend sailing, x, and their distance from shore (in miles), y.
x (hours) y (miles)
1 8
3 24
4 32
9 72
What is the constant of proportionality? Write your answer as a whole number or decimal.
miles per hour

Answer 1

To find the constant of proportionality for the relationship between time spent sailing (x) and distance from shore (y), we can look at the ratio of distance to time:

The constant of proportionality (k) can be calculated using the formula:

\[ k = \frac{y}{x} \]

Let's calculate it for the provided data:

1. For \( x = 1 \) hour and \( y = 8 \) miles: \[ k = \frac{8}{1} = 8 \]

2. For \( x = 3 \) hours and \( y = 24 \) miles: \[ k = \frac{24}{3} = 8 \]

3. For \( x = 4 \) hours and \( y = 32 \) miles: \[ k = \frac{32}{4} = 8 \]

4. For \( x = 9 \) hours and \( y = 72 \) miles: \[ k = \frac{72}{9} = 8 \]

In all cases, the constant of proportionality \( k \) is consistently 8 miles per hour.

Thus, the constant of proportionality is:

\[ \boxed{8} \text{ miles per hour} \]