Time for a rational temperature model to climb

The question asks how long the temperature took to go from $0$ to $5$ degrees. That is a difference in times, so find when each temperature happens and subtract.

With a graphing calculator, graph $y=T(t)$ on $2\le t\le 9$ along with $y=0$ and $y=5$, and find the intersections. On this interval $T$ increases the whole way, from about $-17.78$ at $t=2$ to about $8.98$ at $t=9$, so each temperature happens exactly once.

• $T(t)=0$ at $t\approx 5.420$
• $T(t)=5$ at $t\approx 7.701$

The time it took is $$7.701-5.420\approx 2.280\text{ hours}$$ (Keeping the full calculator values, $7.70072-5.42025=2.28047$, which rounds to 2.280.)

That is choice D.

Each choice

• A. This is the time when the temperature reaches 5 degrees, about 7.70 hours after noon. It answers "when" rather than "how long", so it still needs the starting time subtracted.
• B. This is the time when the temperature is 0 degrees, the start of the rise. The question asks for the length of the rise, which is the gap between this time and the time it hits 5.
• C. This value does not come from the two times at all. Adding it to 5.420 gives about 9.53 hours, beyond the end of the model at $t=9$, so it cannot be the length of the rise. The two intersection times are 2.280 hours apart.
• D (correct). The model gives $T=0$ at $t\approx 5.420$ and $T=5$ at $t\approx 7.701$, and the difference is about 2.280 hours.