Question

A rancher needs to travel from a location on his ranch represented by the point (12, 4) on a coordinate plane to the point (9, 2).

It takes the rancher 10 minutes to travel one mile on horseback. How long will it take him to travel the entire distance between the two points (rounded to the nearest minute)?

Answer 1

To find the time it takes for the rancher to travel from the point (12, 4) to the point (9, 2), we first need to determine the distance between these two points using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Here, \((x_1, y_1) = (12, 4)\) and \((x_2, y_2) = (9, 2)\). Plugging in these coordinates:

\[
d = \sqrt{(9 - 12)^2 + (2 - 4)^2}
\]
\[
d = \sqrt{(-3)^2 + (-2)^2}
\]
\[
d = \sqrt{9 + 4}
\]
\[
d = \sqrt{13}
\]

Next, we need to convert this distance into miles. We can approximate \(\sqrt{13}\):

\[
\sqrt{13} \approx 3.60555 \text{ miles}
\]

Now, knowing that it takes the rancher 10 minutes to travel one mile, we can calculate the time required to travel \(3.60555\) miles:

\[
\text{Time} = \text{Distance} \times \text{Time per mile} = 3.60555 \times 10 \text{ minutes}
\]
\[
\text{Time} \approx 36.0555 \text{ minutes}
\]

Finally, rounding this to the nearest minute, we find that it approximately takes the rancher:

\[
\text{Time} \approx 36 \text{ minutes}
\]

Thus, the answer is:

\[
\boxed{36}
\] minutes.