Problem 34 - Algebra Form and Function

Question

The required cooling capacity, in BTUs, for a room air conditioner is proportional to the area of the room being cooled. A room of 280 square feet requires an air conditioner whose cooling capacity is 5600 BTUs. (a) What is the constant of proportionality, and what are its units? (b) If an air conditioner has a cooling capacity of 10,000 BTUs, how large a room can it cool?

Step-by-Step Solution

Answer

Answer 1: The constant of proportionality (k) is 20, with units of BTUs/square foot. Question 2: What size of room can an air conditioner with a cooling capacity of 10,000 BTUs cool? Answer 2: An air conditioner with a cooling capacity of 10,000 BTUs can cool a room of 500 square feet.

Step 1: Find the constant of proportionality (k)

Since the cooling capacity (C) is proportional to the area of the room (A), we can write the equation as: C = k*A Using the given information (A = 280 square feet and C = 5600 BTUs), we can solve for the constant of proportionality (k): 5600 = k*280 Now, divide both sides by 280 to find the value of k: k = 5600/280 k = 20 The constant of proportionality (k) is 20. Its units are BTUs/square foot.

Step 2: Find the size of a room that an air conditioner with 10,000 BTUs capacity can cool

Given, the cooling capacity of the air conditioner is 10,000 BTUs. We can use the equation C = k*A to find the area of the room (A) it can cool: 10,000 = 20*A Now, divide both sides by 20 to find the value of A: A = 10,000/20 A = 500 The air conditioner with a cooling capacity of 10,000 BTUs can cool a room with an area of 500 square feet. Thus, we have found that: (a) The constant of proportionality (k) is 20, with units of BTUs/square foot. (b) An air conditioner with a cooling capacity of 10,000 BTUs can cool a room of 500 square feet.

Key Concepts

Cooling Capacity

Cooling capacity is a measure of the ability of an air conditioning system to remove heat from a space. It is typically expressed in BTUs (British Thermal Units) per hour.
This value tells you how much heat the air conditioner can remove to keep the room at a comfortable temperature.
When selecting an air conditioner, the cooling capacity should match the size of the room to ensure proper cooling. Understanding Cooling Capacity:

• Measured in BTUs: A higher BTU rating means more cooling power.
• Proportional Relationship: The required cooling capacity increases with the size (area) of the room.

Coolers with higher BTUs can cool larger areas.
Choosing the right cooling capacity ensures energy efficiency and comfort.

Constant of Proportionality

The constant of proportionality is a key concept in understanding relationships between quantities. In this context, it helps relate the cooling capacity of an air conditioner to the area of the room.
The equation used is \[ C = k \times A \]where:

• \( C \) is the cooling capacity in BTUs,
• \( k \) is the constant of proportionality, and
• \( A \) is the area in square feet.

The constant of proportionality here is 20, meaning every square foot requires 20 BTUs of cooling capacity.
This constant helps determine the correct air conditioner size for any given room.

Units Conversion

Units conversion is important when dealing with proportional relationships. It allows for comparing and converting different measured values accurately.
In this situation, the units involved are BTUs and square feet. Why Units Conversion Matters:

• Keeps measurements consistent: Ensures BTUs correlate correctly with the area in square feet.
• Facilitates calculations: Converts measurements into usable units in equations.

Proper conversion ensures that the cooling capacity is effectively utilized, providing adequate cooling for the room size specified.