For a scientific experiment, a physicist must make sure that the temperature of a metal does not get colder than −78 °C. The physicist changes the metal's
temperature at a steady rate of −3 °C per hour.
Let t represent temperature in degrees Celsius. Enter an inequality. Use the fact that the rate of change in temperature times the number of seconds equals the final temperature.
Solve the inequality in the first part. How long will it take the physicist to change the temperature of the metal?
The physicist has to repeat the experiment if the metal gets cooler than −78 °C. How many hours would the physicist have to cool the metal for this to happen?
starting at T, we need after h hours,
T-3h >= -78
16
The inequality can be written as:
Rate of change in temperature * Time = Final temperature
Using the given information,
Rate of change in temperature = -3 °C per hour
Final temperature = -78 °C
The inequality becomes:
-3t ≤ -78
To solve this inequality, we need to divide both sides by -3 to isolate t:
t ≥ -78 / -3
t ≥ 26
Therefore, the inequality is t ≥ 26.
To find out how long it will take the physicist to change the temperature of the metal, we substitute the final temperature into the inequality:
26 hours.
If the physicist wants to cool the metal below -78 °C, the inequality becomes:
-3t < -78
To solve this inequality, we divide both sides by -3:
t > -78 / -3
t > 26
Therefore, the physicist would have to cool the metal for more than 26 hours to reach a temperature lower than -78 °C.
To answer this question, let's break it down step by step:
Step 1: Setting up the inequality
The physicist wants to make sure that the temperature of the metal does not get colder than -78°C. Since the metal's temperature changes at a steady rate of -3°C per hour, we can represent this as an inequality.
Let t represent the temperature in degrees Celsius. The change in temperature can be represented as -3t, where t is the time in hours. The inequality can be written as:
-3t ≥ -78
Step 2: Solving the inequality
To solve the inequality, we need to isolate t. First, divide both sides of the inequality by -3 to get:
t ≤ -78 / -3
Simplifying, we have:
t ≤ 26
Step 3: Interpreting the result
The inequality tells us that the time, t, must be less than or equal to 26 hours for the temperature of the metal to stay above -78°C.
Step 4: How long will it take?
Since the temperature changes at a rate of -3°C per hour, it will take 26 hours to change the temperature of the metal from its initial state to -78°C.
Step 5: How many hours to cool the metal?
If the physicist wants the metal to get colder than -78°C, they would need to cool it for more than 26 hours. The number of additional hours needed would depend on how much colder they want the metal to be.
Remember, this answer assumes a steady rate of -3°C per hour and does not take into account any external factors.