Problem 41 - Thinking Mathematically

Question

A furnace is designed to heat 10,000 cubic feet. Will this furnace be adequate for a 1400 -square-foot house with a 9 -foot ceiling?

Step-by-Step Solution

Answer

No, the furnace will not be adequate for a 1400 square-foot house with a 9-foot ceiling.

Step 1: Calculating the volume of the house

First, to calculate the volume of the house, multiply the square footage of the floor space (1400 square feet) by the height of the ceilings (9 feet). The formula to calculate the volume is: \[ Volume = Area \times height \]. So in our case, the calculation would lead to: \[ 1400 \times 9 = 12,600 \] cubic feet.

Step 2: Compare the house volume with the furnace capacity

Next, we need to compare this volume with the heating capacity of the furnace, which is 10,000 cubic feet. This house volume is larger than the heating capacity of the furnace (12,600 > 10,000) which implies that the furnace is not adequate, i.e., it doesn't have enough capacity to heat the entire volume of the house.

Key Concepts

Volume Calculation

Understanding how to calculate volume is an essential part of solving many real-world problems. In this case, to determine whether a furnace can adequately heat a house, we begin by finding the house's volume. The formula you use for this is

• Volume = Area \(\times\) Height

Here, the area is the floor space of the house, which is given as 1400 square feet. The height corresponds to the ceiling height, standing at 9 feet. By substituting these values into the formula, you get:\[Volume = 1400 \times 9 = 12,600 \]This volume calculation tells us that the house has a total volume of 12,600 cubic feet. This dimension is critical for assessing the heating requirements and ensuring that the house is comfortable year-round.

Furnace Capacity

The concept of furnace capacity deals with the volume of air that a furnace can effectively heat. This is crucial in ensuring that a home environment remains warm and comfortable. A furnace capacity is often specified in cubic feet, which means it can tell you the maximum volume of space it can adequately heat. For the furnace in question, it is designed to heat 10,000 cubic feet. When choosing a furnace, it's important to match its capacity closely with the actual requirements of the space. If the capacity is too low, such as in our example case, the furnace will not perform efficiently. A mismatch might mean:

• The furnace runs constantly, leading to increased energy use and costs.
• Rooms may remain uncomfortably cold, especially those further from the heat source.
• Poor performance can stress the furnace, reducing its lifespan.

Understanding furnace capacity can help ensure you select a system that provides effective heating, consideration important for comfort and efficiency.

Comparative Analysis

In the context of heating a home, comparative analysis involves assessing whether the capabilities of a furnace (or any other system) match the needs of the space. After calculating the volume of the house and knowing the furnace's capacity, the next step is comparing these numbers to ensure a proper match. From our calculations:

• The volume of the house: 12,600 cubic feet
• The furnace's capacity: 10,000 cubic feet

Since 12,600 cubic feet (the volume of the house) is greater than 10,000 cubic feet (the capacity of the furnace), the furnace is not adequate. This inadequacy means the furnace cannot heat the entire space effectively. Through comparative analysis, we can determine the suitability of the furnace to heat the space. It highlights the importance of matching or exceeding the calculated volume against the furnace capacity. Such analysis supports effective decision-making in creating an energy-efficient and comfortable living environment.