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Chapter 2: Problem 34

Solve. For Exercises 29 and \(30,\) the solutions have been started for you. Round all dollar amounts to two decimal places. See Examples 4 and 5. One-foot-square ceiling tiles are sold in packages of \(50 .\) Find how many packages must be bought for a rectangular ceiling 18 feet by 12 feet.

Short Answer

Expert verified
You need 5 packages for the ceiling.

Step by step solution

01

Calculate the Area of the Ceiling

To find the number of packages of ceiling tiles needed, first calculate the area of the ceiling. Since the ceiling is rectangular, use the formula for the area of a rectangle: \[ \text{Area} = \text{length} \times \text{width} \] Substitute the given dimensions (18 feet and 12 feet): \[ \text{Area} = 18 \times 12 \] Calculate the result to find the total area in square feet.
02

Determine the Number of Tiles Needed

Each ceiling tile covers an area of 1 square foot. Therefore, the number of tiles needed equals the total area of the ceiling. From Step 1, you found the area to be 216 square feet. Thus, 216 tiles are needed.
03

Calculate the Number of Packages Required

Ceiling tiles are sold in packages containing 50 tiles each. To find the number of packages needed, divide the total number of tiles by the number of tiles per package:\[ \text{Number of Packages} = \frac{216}{50} \] Complete this division to find a decimal result.
04

Round Up to the Nearest Whole Number

Since the number of packages must be a whole number, and you cannot purchase a fraction of a package, round the result from Step 3 up to the nearest whole number, resulting in the total number of packages required.

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Key Concepts

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

area calculation
When calculating the area of a rectangular shape, such as a ceiling, you use the formula for the area of a rectangle, which is \[ \text{Area} = \text{length} \times \text{width} \]This formula helps you determine how much space the object occupies. If you know the length and width of the rectangle (in this case, 18 feet by 12 feet), you can multiply these two numbers to find the area in square feet.
This is a crucial first step, especially in problems requiring you to cover a surface with materials like tiles or paint. Once you understand how the dimensions relate to the area, other calculations become simpler.
rectangular shapes
Rectangular shapes are common in geometry and everyday life. Think of your ceiling or your dining table; these are often rectangles.
This shape has two parallel and equal sides that are the length and the width. To identify and work with rectangular shapes:
  • Understand that opposite sides of a rectangle are equal.
  • Know that all angles in a rectangle are right angles (90 degrees).
  • Use the designated length and width when solving for area or perimeter, avoiding confusion.
Knowing these attributes aids in solving problems related to rectangular objects efficiently, such as covering the ceiling with tiles.
division rounding
Rounding division results is vital in situations where partial results are impractical. For example, if you have a division result of 4.32, and you need to purchase full packages of ceiling tiles, rounding becomes necessary.
Here's the simple logic behind it:
  • Perform the division needed to solve your problem. Here, dividing the total number of required tiles by the tiles per package.
  • Check if the result is a whole number; if not, round up to the nearest whole number. This ensures you have enough without purchasing too little.
By rounding up, you ensure feasibility and practicality, avoiding the error of not having enough resources.
word problems in math
Word problems require translating text into mathematical expressions and understanding the situation described.
To tackle word problems effectively: - Read the problem fully to understand what is being asked. - Identify the data provided and what needs calculating. - Translate the problem into mathematical equations using relevant formulas. - Perform calculations step by step, applying logical thinking. Word problems often mimic real-world scenarios, which makes them valuable for practical applications. Practice makes you quicker and more precise when converting scenarios into numbers, especially in areas involving calculations like finding how many tile packages you need.

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Most popular questions from this chapter

The expression \(\left|x_{T}-x\right|\) is defined to be the absolute error in \(x\) where \(x_{T}\) is the true value of a quantity and \(x\) is the measured value or value as stored in a computer. If the true value of a quantity is 0.2 and the approximate value stored in a computer is \(\frac{51}{256},\) find the absolute error.

The calorie count of a serving of food can be computed based on its composition of carbohydrate, fat, and protein. The calorie count \(C\) for a serving of food can be computed using the formula \(C=4 h+9 f+4 p,\) where \(h\) is the number of grams of carbohydrate contained in the serving, \(f\) is the number of grams of fat contained in the serving, and p is the number of grams of protein contained in the serving. A serving of raisins contains 130 calories and 31 grams of carbohydrate. If raisins are a fat-free food, how much protein is provided by this serving of raisins?

Solar system distances are so great that units other than miles or kilometers are often used. For example, the astronomical unit \((A U)\) is the average distance between Earth and the sun, or \(92,900,000\) miles. Use this information to convert each planet's distance in miles from the sun to astronomical units. Round to three decimal places. The planet Mercury's AU from the sun has been completed for you. (Source: National Space Science Data Center). $$ \begin{array}{|c|c|c|} \hline \text { Planet } & {\text { Miles from the Sun }} & {\text { AUfrom the Sun }} \\ \hline \text { Mercury } & {36 \text { million }} & {0.388} \\ \hline \end{array} $$ Neptune 2793 million

Solve each inequality. Graph the solution set and write it in interval notation. $$ |4+9 x| \geq-6 $$

Write an absolute value inequality representing all numbers \(x\) whose distance from 0 is greater than 4 units.
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