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Chapter 4: Problem 20

A homeowner would like to spread shredded bark (mulch) over her flowerbeds. She has three flowerbeds measuring \(25 \mathrm{ft}\) by \(3 \mathrm{ft}, 15 \mathrm{ft}\) by \(4 \mathrm{ft},\) and \(30 \mathrm{ft}\) by \(1.5 \mathrm{ft}\). The recommended depth for the mulch is 4 inches, and the shredded bark costs \(\$ 27.00\) per one cubic yard. How much will it cost to cover all of the flowerbeds with shredded bark? (Note: You cannot buy a portion of a cubic yard of mulch.)

Short Answer

Expert verified
The cost will be $81.

Step by step solution

01

Convert Depth into Feet

The depth of the mulch is given in inches, convert it to feet since the other dimensions are in feet. There are 12 inches in a foot, so the depth in feet is \[ 4 \text{ inches} = \frac{4}{12} \text{ feet} = \frac{1}{3} \text{ feet} \]
02

Calculate the Volume of Each Flowerbed

Use the formula for volume \( V = \text{length} \times \text{width} \times \text{depth} \). Calculate the volume for each flowerbed: \[ \text{First flowerbed}: 25 \text{ ft} \times 3 \text{ ft} \times \frac{1}{3} \text{ ft} = 25 \text{ ft} \times 1 \text{ ft} = 25 \text{ cubic feet} \ \text{Second flowerbed}: 15 \text{ ft} \times 4 \text{ ft} \times \frac{1}{3} \text{ ft} = 15 \text{ ft} \times 1.333.... \text{ ft} \approx 20 \text{ cubic feet} \ \text{Third flowerbed}: 30 \text{ ft} \times 1.5 \text{ ft} \times \frac{1}{3} \text{ ft} = 30 \text{ ft} \times 0.5 \text{ ft} = 15 \text{ cubic feet} \]
03

Calculate Total Volume

Sum the volumes of the three flowerbeds: \[ 25 \text{ cubic feet} + 20 \text{ cubic feet} + 15 \text{ cubic feet} = 60 \text{ cubic feet} \]
04

Convert Cubic Feet to Cubic Yards

There are 27 cubic feet in a cubic yard. Convert the total volume to cubic yards: \[ \frac{60 \text{ cubic feet}}{27 \text{ cubic feet per cubic yard}} \approx 2.22 \text{ cubic yards} \]
05

Calculate Cost

Since mulch is sold by the cubic yard and we cannot buy a fraction of a yard, round up to the nearest whole number, which is 3 cubic yards. The cost is \[ 3 \text{ cubic yards} \times 27 \text{ dollars per cubic yard} = 81 \text{ dollars} \]

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Key Concepts

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

unit conversion
Unit conversion is a critical step in any calculation involving different measurement units. It's essential for ensuring consistency across all computations. In our exercise, the depth of the mulch is given in inches, but the measurements of the flowerbeds are in feet. To align all measurements, we convert inches to feet. Remember, there are 12 inches in a foot.

For example, converting 4 inches to feet involves dividing by 12: \[\frac{4 \text{ inches}}{12} = \frac{1}{3} \text{ feet} \]
This conversion simplifies further calculations and avoids errors that may arise from mixed units.
volume formula
The volume of a three-dimensional space is calculated using the formula: \[ V = \text{length} \times \text{width} \times \text{depth} \] In our case, we apply this formula to each flowerbed to determine the amount of mulch needed.

Three flowerbeds require separate volume calculations:
  • First flowerbed: 25 ft by 3 ft by \[ \frac{1}{3} \text{ ft} \]
  • Second flowerbed: 15 ft by 4 ft by \[ \frac{1}{3} \text{ ft} \]
  • Third flowerbed: 30 ft by 1.5 ft by \[ \frac{1}{3} \text{ ft} \]
Multiplying these values, we find: \[ 25 \text{ cubic feet, } \approx 20 \text{ cubic feet, and } 15 \text{ cubic feet} \] Summing these volumes provides the total amount of mulch required: \[ 25 + 20 + 15 = 60 \text{ cubic feet} \]
rounding in calculations
Rounding ensures that we present our final answers in a practical and applicable form, especially in real-world scenarios where measurements and purchases can't always be precise. Sometimes, this involves rounding up to avoid shortages.

In our example, the total volume of mulch needed is converted from cubic feet to cubic yards, which results in approximately 2.22 cubic yards. Since mulch cannot be purchased in a fraction of a yard, we round 2.22 cubic yards up to 3 cubic yards. This ensures that the homeowner has enough mulch to cover her flowerbeds adequately.
cost calculation
Once the volume is determined and rounded, cost calculation becomes straightforward. We multiply the total number of whole cubic yards by the price per cubic yard.

In our case, the mulch costs $27.00 per cubic yard. Therefore, for 3 cubic yards, the cost calculation is: \[ 3 \text{ cubic yards} \times 27 \text{ dollars per cubic yard} = 81 \text{ dollars} \]
This simple multiplication gives the total cost of the mulch needed to cover the flowerbeds. Always ensure to purchase a whole number of units when dealing with bulk materials to meet project requirements without running short.

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Most popular questions from this chapter

Determine the order-of-magnitude difference in the sizes of the radii for: a. The solar system \(\left(10^{12}\right.\) meters) compared with Earth \(\left(10^{7}\right.\) meters) b. Protons \(\left(10^{-15}\right.\) meter) compared with the Milky Way \(\left(10^{21}\right.\) meters) c. Atoms \(\left(10^{-10}\right.\) meter) compared with neutrons \(\left(10^{-15}\right.\) meter)

A TV signal traveling at the speed of light takes about \(8 \cdot 10^{-5}\) second to travel 15 miles. How long would it take the signal to travel a distance of 3000 miles?

An electron weighs about \(10^{-27}\) gram, and a raindrop weighs about \(10^{-3}\) gram. How many times heavier is a raindrop than an electron? How many times lighter is an electron than a raindrop? What is the order-of- magnitude difference?

Radio waves, sent from a broadcast station and picked up by the antenna of your radio, are a form of electromagnetic (EM) radiation, as are microwaves, X-rays, and visible, infrared, and ultraviolet light. They all travel at the speed of light. Electromagnetic radiation can be thought of as oscillations like the vibrating strings of a violin or guitar or like ocean swells that have crests and troughs. The distance between the crest or peak of one wave and the next is called the wavelength. The number of times a wave crests per minute, or per second for fast-oscillating waves, is called its frequency. Wavelength and frequency are inversely proportional: the longer the wavelength, the lower the frequency, and vice versa-the faster the oscillation, the shorter the wavelength. For radio waves and other \(\mathrm{EM}\), the number of oscillations per second of a wave is measured in hertz, after the German scientist who first demonstrated that electrical waves could transmit information across space. One cycle or oscillation per second equals 1 hertz \((\mathrm{Hz})\). For the following exercise you may want to find an old radio or look on a stereo tuner at the AM and FM radio bands. You may see the notation \(\mathrm{kHz}\) beside the AM band and MHz beside the FM band. AM radio waves oscillate at frequencies measured in the kilohertz range, and FM radio waves oscillate at frequencies measured in the megahertz range. a. The Boston FM rock station WBCN transmits at \(104.1 \mathrm{MHz}\). Write its frequency in hertz using scientific notation. b. The Boston AM radio news station WBZ broadcasts at 1030 \(\mathrm{kHz}\). Write its frequency in hertz using scientific notation. The wavelength \(\lambda\) (Greek lambda) in meters and frequency \(\mu\) (Greek mu) in oscillations per second are related by the formula \(\lambda=\frac{c}{\mu}\) where \(c\) is the speed of light in meters per second. c. Estimate the wavelength of the WBCN FM radio transmission. d. Estimate the wavelength of the WBZ AM radio transmission. e. Compare your answers in parts (c) and (d), using orders of magnitude, with the length of a football field (approximately 100 meters).

Change each number into scientific notation, then perform the indicated calculation without a calculator. a. A \(\$ 600,000\) lottery jackpot is divided among 300 people. What are the winnings per person? b. A total of 2500 megawatts are used over 500 hours. What is the rate in watts per hour? c. If there were 6 million births in 30 years, what is the birth rate per year?
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