Question

Raj has put a soup into the freezer to cool down. The soup's temperature started at 170 degrees F and decreased at a rate of 30 degrees F per hour. Let h represent the number of hours the soup stays in the freezer.

Give the number of hours that Raj could leave the soup in the freezer that makes both inequalities true.

Answer 1

To find the number of hours (h) that Raj could leave the soup in the freezer while satisfying both inequalities, we'll consider the given conditions:

The initial temperature of the soup is 170 degrees F.

The temperature decreases at a rate of 30 degrees F per hour.

So, after h hours in the freezer, the temperature of the soup can be represented as:

Temperature = Initial Temperature - Rate of Decrease * Number of Hours

Temperature = 170 - 30h

Now, let's set up the inequalities:

a) The temperature should be less than or equal to 110 degrees F:

170 - 30h ≤ 110

b) The temperature should be greater than or equal to 80 degrees F:

170 - 30h ≥ 80

Now, solve each inequality separately:

For the first inequality (170 - 30h ≤ 110):

170 - 30h ≤ 110

Subtract 170 from both sides:

-30h ≤ -60

Divide by -30 (remember to reverse the inequality because you're dividing by a negative number):

h ≥ 2

For the second inequality (170 - 30h ≥ 80):

170 - 30h ≥ 80

Subtract 170 from both sides:

-30h ≥ -90

Divide by -30 (again, reverse the inequality):

h ≤ 3

So, to satisfy both inequalities, the number of hours (h) that Raj could leave the soup in the freezer is between 2 and 3 hours, inclusive. In other words, 2 hours ≤ h ≤ 3 hours.

Answer 2

Te = Ts - (30F/hr)*(h)

Explanation:

h is the hours the soup stays in the freezer.

Let Ts be the starting temperature, and Te the ending temperature.

We are told that the rate of cooling is -30F/hr.

We can write Te = Ts - (30F/hr)*(h)

For the example, Ts is 170F

Te = 170F - (30F/hr)*(h)

The question "Give the number of hours that Raj could leave the soup in the freezer that makes both inequalities true." seems to be missing some information. What inequalities? Is there a target temperature? If a target temperature is given, use the above equation to find the number of hours to reach that temperature. For example, if the target temperature is 150F:

Te = 170F - (30F/hr)*(h)

140F = 170F - (30F/hr)*(h)

- 30F = - (30F/h)*h

h = 1 hour