Question

The Inuit people in the arctic regions of North America sometimes build igloos of snow blocks. During winter months, the temperatures can be as low as \(-58^{\circ} \mathrm{F}\). A small heater can raise the internal temperature of a snow-block igloo by \(108^{\circ} \mathrm{F}\). What would the temperature be inside the igloo? (PICTURE CANT COPY)

Short answer

Answer: The final temperature inside the igloo is 50°F.

Step-by-step solution

Identify and write down the given information

We are given the original outdoor temperature as \(-58^{\circ}\mathrm{F}\) and the increase in temperature due to the heater as \(108^{\circ}\mathrm{F}\).

Add the increase in temperature to the original temperature

To find the final temperature inside the igloo, we simply add the increase in temperature to the original temperature: Final temperature \( = -58^{\circ}\mathrm{F} + 108^{\circ}\mathrm{F}\)

Calculating the final temperature

Perform the addition: Final temperature \( = 50^{\circ}\mathrm{F}\)

State the final answer

The temperature inside the igloo would be \(50^{\circ}\mathrm{F}\).

Key concepts

Adding Negative and Positive Numbers

Understanding how to add negative and positive numbers is essential in prealgebra and is a key part of developing mathematical literacy. In the context of temperature, think of negative numbers as representing cold and positive numbers as representing warmth. Adding a positive number to a negative one is like adding heat to a cold place—it will raise the temperature.

For example, if the temperature is at \( -58^\circ \mathrm{F} \) and we introduce a heat source that provides \( 108^\circ \mathrm{F} \), we are essentially moving up 108 units on the number line from -58. Visualizing a number line can be incredibly helpful; negative numbers are to the left, positive numbers to the right, and the more we move to the right, the more we offset the negative value with positive units.

Here's how you can visualize this addition: Start at -58 and count 108 steps to the right. Each step brings you closer to zero (which is the point of neutrality in temperature) and then beyond it into the positive range. In the igloo example, the internal temperature eventually reaches \( 50^\circ \mathrm{F} \) after adding 108 steps, or degrees in this case, from -58.

Real-World Math Application

Applying what we've learned in math to real-world situations solidifies our understanding and appreciation of the subject. The example of the Inuit people using a heater inside an igloo showcases a practical application of adding negative and positive numbers. Whether it's in calculating temperatures, managing finances, or measuring elevation changes in geography, the principles of working with integers remain consistent.

In our igloo example, this math application gave us a way to predict how effective a heater can be in the extreme cold. Such calculations could be crucial for survival and comfort in arctic environments, showcasing that math is not just an academic exercise but a critical tool for real-life problem-solving. Understanding these principles allows us to make informed decisions, whether that's preparing for a cold night in the arctic or adjusting the thermostat at home.

Temperature Conversion

While the example given works directly with Fahrenheit temperatures, a skill often needed in math and science is the ability to convert temperatures between different systems, like Celsius and Kelvin. Temperature conversion formulas are mathematical expressions that allow us to translate one unit of temperature to another.

For instance, to convert Fahrenheit to Celsius, we use the formula \( \frac{(F - 32) \times 5}{9} = C \), where \( F \) represents degrees Fahrenheit, and \( C \) represents degrees Celsius. Conversely, if you wish to go from Celsius to Fahrenheit, you would use the formula \( \frac{(C \times 9)}{5} + 32 = F \).

Using the original exercise's final temperature of \( 50^\circ \mathrm{F} \), if we wanted to convert this to Celsius, we would plug in 50 for \( F \) in our conversion formula to get \( \frac{(50 - 32) \times 5}{9} = \frac{18 \times 5}{9} = 10^\circ \mathrm{C} \). Thus, math not only helps us understand changes in temperature but also equips us to communicate this information across different temperature scales, broadening our ability to work internationally or in various scientific contexts.