Question

The temperature in Boone, NC started -14 degrees C and rose 2 degrees every hour. The temperature in Monroe, NC started at -2 degrees C and rose 1 degree every 2 hours.



After how many hours will the temperature be the same in both cities?

Answer 1

To find out when the temperatures in Boone, NC, and Monroe, NC will be the same, we can set up equations for the temperatures of both cities as functions of time in hours.

Let \( t \) be the number of hours that have passed.

1. For Boone, NC:
- Initial temperature: \(-14\) degrees C.
- Temperature increase: \(2\) degrees every hour.
- Temperature after \( t \) hours:
\[
T_B(t) = -14 + 2t
\]

2. For Monroe, NC:
- Initial temperature: \(-2\) degrees C.
- Temperature increase: \(1\) degree every \(2\) hours, which is \(0.5\) degrees every hour.
- Temperature after \( t \) hours:
\[
T_M(t) = -2 + 0.5t
\]

Now we want to find \( t \) when the temperatures are equal, so we set \( T_B(t) \) equal to \( T_M(t) \):
\[
-14 + 2t = -2 + 0.5t
\]

To solve for \( t \), we first rearrange the equation:
1. Move all terms involving \( t \) to one side and constant terms to the other side:
\[
2t - 0.5t = -2 + 14
\]
\[
1.5t = 12
\]

2. Divide both sides by \( 1.5 \):
\[
t = \frac{12}{1.5} = 8
\]

Thus, after \( \boxed{8} \) hours, the temperatures in Boone, NC, and Monroe, NC will be the same.

Let's verify:
- Temperature in Boone after 8 hours:
\[
T_B(8) = -14 + 2(8) = -14 + 16 = 2 \text{ degrees C}
\]

- Temperature in Monroe after 8 hours:
\[
T_M(8) = -2 + 0.5(8) = -2 + 4 = 2 \text{ degrees C}
\]

Both temperatures equal \( 2 \) degrees C after \( 8 \) hours, confirming that the solution is correct.