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Chapter 4: Problem 19

A hemispherical dome with a 50 -foot radius will be given a coat of paint 0.01 inch thick. The contractor for the job wants to estimate the number of gallons of paint that will by needed. Use a differential to obtain an estimate. (There are 231 cubic inches in a gallon.)

Short Answer

Expert verified
The estimated number of gallons of paint that will be needed can be calculated using the steps above.

Step by step solution

01

Calculating the original volume

First, the volume of the hemisphere needs to be calculated. The volume of a sphere is given by the formula \((4/3) \pi r^3\), so the volume of a hemisphere would be half of that, \((2/3) \pi r^3\). Given that the radius \(r\) is 50 feet, which is \(50 * 12 = 600\) inches, the initial volume \(V_{1}\) of the hemisphere is \((2/3) \pi (600)^3\) cubic inches.
02

Calculating the new volume

Next, the new volume of the hemisphere, including the layer of paint, needs to be calculated. Given that the paint layer is 0.01 inches thick, the new radius \(r'\) is \(600 + 0.01 = 600.01\) inches. So, the new volume \(V_{2}\) of the hemisphere is \((2/3) \pi (600.01)^3\)
03

Calculating the volume of the paint

The volume of paint required is the difference between the new volume and the original volume. This can be calculated as \(V_{2} - V_{1}\). Calculate this difference to find the answer in cubic inches.
04

Converting cubic inches to gallons

Given that there are 231 cubic inches in a gallon, you can convert the volume of paint required to gallons by dividing the volume in cubic inches by 231. This will give you the final answer in gallons.

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Key Concepts

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

Volume Calculation
When dealing with shapes like hemispheres, calculating volume is a fundamental step. In geometry, a sphere's volume is calculated using the formula: \((4/3) \pi r^3\). Because a hemisphere is exactly half of a sphere, its volume is half of that, which is given by the formula: \((2/3) \pi r^3\).

This formula gives us the total amount of space inside the hemisphere. For our exercise, we start by plugging the radius into this formula to find the original volume of the dome. Calculating this correctly sets the groundwork for further steps in the problem, such as estimating the paint needed when the radius changes. Always ensure units are consistent to avoid errors in calculations.
Hemisphere
A hemisphere is essentially half of a sphere. This particular shape is prevalent in various real-world applications, from domes in architecture to earth science models. Understanding the hemisphere's properties helps us derive its volume and surface area, which are crucial for tasks like painting.

The dome in our exercise is a classic example. With a given radius, you need to calculate the changes in the hemisphere when a layer, such as paint, is added. When dealing with problems involving hemispheres, always remember that calculations are half of those for a full sphere, whether it's for volume or surface area.
Radius Conversion
Converting units of measurement for the radius is essential in accurate scientific calculations. In our exercise, the radius of the dome is initially given in feet. However, since the paint thickness is measured in inches, we must first convert feet to inches to ensure that all measurements are consistent.

The conversion from feet to inches is straightforward: multiply the number of feet by 12. Hence, a 50-foot radius becomes a 600-inch radius. Keeping units uniform simplifies calculations and prevents potential mistakes, especially when dealing with differential calculus for volume changes.
Unit Conversion
Unit conversion is critical when working with differential calculus in practical applications like estimating paint usage. In our exercise, we're working with volume measurements both in cubic inches and gallons.

To convert the volume of paint from cubic inches to gallons, it's important to know that one gallon equals 231 cubic inches. After determining the paint's volume in cubic inches, simply divide this number by 231 to convert it to gallons. This conversion allows us to assess the physical, practical requirements (like buying paint) from mathematical calculations.

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Most popular questions from this chapter

The Hot wheels Rent-A-Car Company derives an average net profit of 12 dollar per customer if it services 50 customers or fewer. If it services more than 50 customers, then the average net profit is decreased by 6 cents for each customer over 50 . What number of customers produces the greatest total net profit for the company?

Use a differential to estimate the change in the volume of a cube caused by an increase \(h\) in the length of each side. Interpret geometrically the error of your estimate \(\Delta V-d \mathcal{V}\).

Neglect air resistance. For the numerical calculations, take \(g\) as 32 feet per second \(\mathrm{p}\) er second or as 9.8 meters per second per second. Supplies are dropped from a stationary helicopter and seconds later hit the ground at 98 meters per second. How high was the helicopter?

An airplane is flying at constant speed and altitude on a line that will take it directly over a radar station on the ground. At the instant the plane is 12 miles from the station, it is noted that the plane's angle of elevation is \(30^{\circ}\) and is increasing at the rate of \(0.5^{\circ}\) per second. Give the speed of the plane in miles per hour.

Water flows from a faucet into a hemispherical basin 14 inches in diameter at the rate of 2 cubic inches per second. How fast does the water rise (a) when the water is exactly halfway to the top? (b) just as it runs over? (The volume of a spherical segment is given by \(\pi r h^{2}-\frac{1}{3} \pi h^{3}\) where \(r\) is the radius of the sphere and \(h\) is the depth of the segment.)
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