Question

1. on a winter morning, the temperature before sunrise was -10°f. the temperature then rose by 1° each hour for 7 hours before dropping by 2° each hour for 3 hours. what was the temperature, in degrees fahrenheit, after 10 hours?

Explanation:

Step1: Calculate temperature after rising

Initial temp: \(-10^\circ\text{F}\). Rises \(1^\circ\) per hour for 7 hours. So change: \(1\times7 = 7\). New temp: \(-10 + 7 = -3^\circ\text{F}\).

Step2: Calculate temperature after dropping

Drops \(2^\circ\) per hour for 3 hours. Change: \(2\times3 = 6\) (drop means subtract). New temp: \(-3 - 6 = -9^\circ\text{F}\)? Wait, wait, original work had errors. Wait, let's redo. Wait, rising: \(-10 + 7\times1 = -3\). Then dropping: \(2^\circ\) per hour for 3 hours, so total drop is \(2\times3 = 6\). So \(-3 - 6 = -9\)? Wait, but the original handwritten had \(-10 +7 -3\) (wrong, should be \(-10 +7 - (2\times3)\)). Wait, no: rising 7 hours: +7, then dropping 3 hours: -2*3. So \(-10 + 7 - 6 = -9\). Wait, the handwritten had \(-3 -3 = -6\) which is wrong. Let's do correctly.

Wait, let's start over. Initial: \(-10\). First 7 hours: +1 each hour: total rise \(7\times1 = 7\). So temp after 7 hours: \(-10 + 7 = -3\). Then next 3 hours: drops 2 each hour: total drop \(3\times2 = 6\). So temp after 10 hours: \(-3 - 6 = -9\). Wait, but maybe the original problem's handwritten was wrong. Wait, the problem says "dropping by 2° each hour for 3 hours", so total drop is 2*3=6. So correct calculation: \(-10 + 7 - 6 = -9\). But let's check the steps again.

Wait, maybe I misread. Wait, the problem: "rose by 1° each hour for 7 hours" → +7. Then "dropping by 2° each hour for 3 hours" → -2*3 = -6. So total: -10 +7 -6 = -9.

But let's follow the correct steps:

Step1: Temperature after rising phase

Initial temperature: \(-10^\circ\text{F}\).
Rise per hour: \(1^\circ\), hours: 7.
Total rise: \(1 \times 7 = 7^\circ\).
Temperature after rising: \(-10 + 7 = -3^\circ\text{F}\).

Step2: Temperature after dropping phase

Drop per hour: \(2^\circ\), hours: 3.
Total drop: \(2 \times 3 = 6^\circ\).
Temperature after dropping: \(-3 - 6 = -9^\circ\text{F}\).

Answer:

\(-9^\circ\text{F}\)