Problem 103 - Elementary and Intermediate Algebra

Question

World's Coldest lce Cream. Dippin' Dots is an ice cream snack that was invented by Curt Jones in \(1987 .\) The tiny multicolored beads are created by flash freezing ice cream mix in liquid nitrogen at a temperature of \(-355^{\circ} \mathrm{F} .\) When they come out of the processor, they are stored at a temperature of \(-40^{\circ} \mathrm{F} .\) Find the change in temperature of Dippin' Dots from production to storage.

Step-by-Step Solution

Answer

The temperature increase from production to storage is \(315^{\circ} \mathrm{F} \).

Step 1: Understand the Temperatures

Identify the two given temperatures: the production temperature is \(-355^{\circ} \mathrm{F} \) and the storage temperature is \(-40^{\circ} \mathrm{F} \).

Step 2: Use the Formula for Change in Temperature

Use the formula: change in temperature = final temperature - initial temperature. In this context, the final temperature is the storage temperature, and the initial temperature is the production temperature.

Step 3: Substitute the Values

Substitute the given values into the formula: \[ \text{Change in Temperature} = (-40) - (-355) \]

Step 4: Simplify the Expression

Simplify the expression based on the order of operations (PEMDAS/BODMAS):\((-40) - (-355)\) becomes \(-40 + 355\) because subtracting a negative is the same as adding.\[ -40 + 355 = 315 \]

Step 5: Interpret the Result

The positive result indicates an increase in temperature. Therefore, from production to storage, the temperature of Dippin’ Dots increases by \(315^{\circ} \mathrm{F} \).

Key Concepts

Understanding Temperature Change

When we talk about temperature change, we are focusing on how much the temperature shifts from one point to another. It is not enough to know the numbers; understanding the process is key.
For example, in our case with Dippin' Dots, we start with a production temperature of -355°F and move to a storage temperature of -40°F. The question to solve here is: "How much warmer is the storage compared to when it was produced?"

To find out, we use the formula:

• Change in Temperature = Final Temperature - Initial Temperature.

Here, the initial temperature is the cooler production temperature and the final temperature is the warmer storage temperature. Once we substitute these values into the formula, we can determine the change. This gives us a clear picture of how much the temperature has increased.

Subtraction of Integers

Subtraction with integers can be tricky, especially with negative numbers involved. Yet, knowing how to handle them is crucial, particularly with temperature changes as illustrated in our exercise.
The original problem asked us to subtract (-355) from (-40):

• The formula looks like this: [-40 - (-355)].

To effectively solve this, we apply a simple rule: subtracting a negative number is the same as adding the absolute value of that number. Hence, (-40 - -355) becomes (-40 + 355). It's like a double negative, which in effect turns into a positive.

This transformation is at the heart of the calculation, simplifying it and ensuring that we arrive at the correct result. This critical step leads us to find that (-40 + 355) equals 315.

Simplifying with PEMDAS/BODMAS

When simplifying expressions, especially those involving multiple steps, the PEMDAS/BODMAS rule is your best friend. It stands for:

• Parentheses/Brackets
• Exponents/Orders
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our current solution, primarily subtraction and addition play a role. Upon applying the subtraction of integers rule, we transformed (-40 - -355) into (-40 + 355).

Once in this form, it becomes a straightforward addition problem. By following PEMDAS/BODMAS here effectively, we ensure that all operations are performed correctly and in the right sequence. This approach is crucial in maintaining mathematical precision, ultimately guiding us to conclude that the temperature change in Dippin' Dots from production to storage is indeed 315°F.