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Chapter 1: Problem 54

The description for a certain brand of house paint claims a coverage of \(460 \mathrm{ft}^{2} /\) gal. (a) Express this quantity in square meters per liter. (b) Express this quantity in an SI unit (see Appendices A and D. (c) What is the inverse of the original quantity, and (d) what is its physical significance?

Short Answer

Expert verified
(a) 11.285 m虏/L, (b) 11285 m鈦宦�, (c) 0.0022 gal/ft虏, (d) Paint needed per square foot.

Step by step solution

01

Convert Square Feet to Square Meters

First, let's convert the area given in square feet to square meters. The conversion factor is \(1 \text{ ft}^2 = 0.092903 \text{ m}^2\). So, we have:\[460 \text{ ft}^2 = 460 \times 0.092903 \text{ m}^2 = 42.73638 \text{ m}^2.\]
02

Convert Gallons to Liters

Next, convert gallons to liters. The conversion factor is \(1 \text{ gal} = 3.78541 \text{ L}\). So, the original quantity is:\[1 \text{ gal} = 3.78541 \text{ L}.\]
03

Calculate Square Meters per Liter

Combine the previous steps to find the coverage in square meters per liter:\[ rac{42.73638 \text{ m}^2}{3.78541 \text{ L}} = 11.285 \text{ m}^2/\text{L}.\]
04

Express the Quantity in an SI Unit

The SI unit for area per volume is \(\text{m}^2/\text{m}^3\), which simplifies to \(\text{m}^{-1}\). Since we have calculated \(11.285 \text{ m}^2/\text{L}\) and \(1 \text{ L} = 10^{-3} \text{ m}^3\), the SI unit expression is:\[11.285 \times 10^{3} \text{ m}^{-1} = 11285 \text{ m}^{-1}.\]
05

Calculate the Inverse Quantity

The inverse of the original coverage is gallons per square foot. Therefore:\[ rac{1}{460} \text{ gal/ft}^2 = 0.002173913 \text{ gal/ft}^2.\]
06

Interpret the Physical Significance

The inverse represents the amount of paint in gallons needed to cover one square foot. It helps determine how much paint is required for small areas.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Meters to Liters
In order to convert coverage from square feet per gallon to square meters per liter, we need to tackle both the area and the volume.
First, we convert square feet to square meters. The conversion factor is simple;
  • 1 square foot is equivalent to 0.092903 square meters. This is achieved by multiplying the given area in square feet by this conversion factor.
For our exercise, the area is 460 square feet, which translates to approximately 42.73638 square meters when using the factor.
Next is the volume conversion鈥攆rom gallons to liters.
  • 1 gallon equals 3.78541 liters. By multiplying the volume in gallons by this conversion factor, we obtain the equivalent volume in liters.
The coverage amount in our example becomes 3.78541 liters.
Finally, achieving the desired units, we divide the calculated square meters by the liters;
  • 42.73638 square meters divided by 3.78541 liters results in approximately 11.285 square meters per liter.
This tells us how much area can be painted with one liter of paint.
SI Units
The International System of Units, or SI units, is the standardized method used globally for scientific and technical measurements.
In our conversion task, we need to express coverage in SI units.
Here, the SI unit for "area per volume"
  • is described through square meters per cubic meter (\(\text{m}^2/\text{m}^3\)),
which simplifies to just \(\text{m}^{-1}\)
because dividing area by volume means effectively reducing dimensions from two to one.
In the problem example, we already know the conversion to meters per liter:
  • 11.285 square meters per liter turns into
  • 11.285 times 1000 meters per centimeter cube (since 1 liter is \(10^{-3}\) cubic meters).
This results in expressing coverage as 11,285 meters \((\text{m}^{-1})\).
It's a good reminder of how changing measurement scales can adjust big numbers to more convenient forms, and why using SI units standardizes these calculations globally.
Inverse Physical Significance
Understanding the inverse physical significance of our measurements can deepen our comprehension of what the values represent.
When we talk about inverting the original coverage (as given in the exercise),
  • we simply reverse the proportion to express the paint's volume requirement per unit of area.
For the exercise, converting "gallons per square foot," gives
  • a value of 0.002173913 gallons per square foot.
This clearly translates into the number of gallons needed to cover a single square foot of surface.
Such inversions have practical applications;
for instance, when you're determining how much paint is needed for small spaces,
  • the value advises on meticulous allocation of paint resources based on surface areas you're planning to cover.
Through these conversions and inversions, you gain a versatile understanding of both coverage efficiency and resource management.

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Most popular questions from this chapter

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