Problem 29
Question
A company has contracted to manufacture 10,000 closed wooden crates having dimensions \(3 \mathrm{ft}, 4 \mathrm{ft}\), and \(5 \mathrm{ft}\). The cost of the wood to be used is \(5 \notin\) per square foot. If the machines that are used to cut the pieces of wood have a possible error of \(0.05 \mathrm{ft}\) in each dimension, find approximately, by using the total differential, the greatest possible error in the estimate of the cost of the wood.
Step-by-Step Solution
Verified Answer
The greatest possible error in the estimate of the cost of the wood is 12 cents per crate.
1Step 1 - Define the Surface Area of the Crate
The surface area of the rectangular crate is given by: \[ A = 2(lw + lh + wh) \]Substituting the dimensions: \[ A = 2(3 \times 4 + 3 \times 5 + 4 \times 5) \]Calculate the numeric value.\[ A = 2(12 + 15 + 20) = 2 \times 47 = 94 \text{ square feet} \]
2Step 2 - Define the Total Cost of Wood Formula
The total cost of wood can be written as:\[ C = 5 \times 94 = 470 \text{ cents per crate} \]
3Step 3 - Express the Surface Area Function
The surface area function based on dimensions \( l, w, h \) is: \[ A(l, w, h) = 2(lw + lh + wh) \]
4Step 4 - Find the Partial Derivatives
Compute the partial derivatives of the surface area with respect to \( l, w, h \):\[ \frac{\text{∂A}}{\text{∂l}} = 2(w + h) \]\[ \frac{\text{∂A}}{\text{∂w}} = 2(l + h) \]\[ \frac{\text{∂A}}{\text{∂h}} = 2(l + w) \]Substitute the values for \( l = 3 \text{ ft}, w = 4 \text{ ft}, h = 5 \text{ ft} \):\[ \frac{\text{∂A}}{\text{∂l}} = 2(4 + 5) = 18 \text{ square feet per ft} \]\[ \frac{\text{∂A}}{\text{∂w}} = 2(3 + 5) = 16 \text{ square feet per ft} \]\[ \frac{\text{∂A}}{\text{∂h}} = 2(3 + 4) = 14 \text{ square feet per ft} \]
5Step 5 - Calculate the Total Differential
The total differential of the surface area \( dA \) is:\[ dA = \frac{\text{∂A}}{\text{∂l}} \text{d}l + \frac{\text{∂A}}{\text{∂w}} \text{d}w + \frac{\text{∂A}}{\text{∂h}} \text{d}h \]Given the error \( \text{d}l = \text{d}w = \text{d}h = 0.05 \text{ ft} \):\[ dA = 18 \times 0.05 + 16 \times 0.05 + 14 \times 0.05 \]Simplify this to find \( dA \).\[ dA = 0.9 + 0.8 + 0.7 = 2.4 \text{ square feet} \]
6Step 6 - Estimate the Greatest Possible Error in Cost
The greatest possible error in the cost can be estimated by multiplying the error in surface area by the cost per square foot:\[ \text{Error in Cost} = 5 \times 2.4 = 12 \text{ cents per crate} \]
Key Concepts
Total DifferentialPartial DerivativesSurface Area CalculationError Propagation
Total Differential
The total differential is a concept in calculus used to approximate the change in a function due to small changes in its variables. It is particularly useful for error estimation in measurements. For a function of several variables like the surface area of a crate, the total differential gives the combined effect of small changes in each of the dimensions.
Mathematically, for a function \(f(x, y, z)\), the total differential \(df\) is given by:
\[ df = \frac{\text{∂f}}{\text{∂x}} dx + \frac{\text{∂f}}{\text{∂y}} dy + \frac{\text{∂f}}{\text{∂z}} dz \]
In our exercise, we use this differential to estimate the error in the surface area and ultimately, the cost of the wood needed for making crates. The key understanding here is that even small measurement errors can propagate and lead to significant errors in derived quantities.
Mathematically, for a function \(f(x, y, z)\), the total differential \(df\) is given by:
\[ df = \frac{\text{∂f}}{\text{∂x}} dx + \frac{\text{∂f}}{\text{∂y}} dy + \frac{\text{∂f}}{\text{∂z}} dz \]
In our exercise, we use this differential to estimate the error in the surface area and ultimately, the cost of the wood needed for making crates. The key understanding here is that even small measurement errors can propagate and lead to significant errors in derived quantities.
Partial Derivatives
Partial derivatives are used to measure how a function changes as its variables change individually, while keeping the other variables constant. They are fundamental in the calculation of the total differential.
For a function \(A(l, w, h) = 2(lw + lh + wh)\), the partial derivatives with respect to each variable are:
\[ \frac{\text{∂A}}{\text{∂l}} = 2(w + h) \]
\[ \frac{\text{∂A}}{\text{∂w}} = 2(l + h) \]
\[ \frac{\text{∂A}}{\text{∂h}} = 2(l + w) \]
This shows how each dimension (length, width, height) individually affects the surface area. By substituting specific values for the dimensions of the crate (\(l = 3 \text{ ft}, w = 4 \text{ ft}, h = 5 \text{ ft}\)), we get the specific partial derivatives needed to compute the total differential.
For a function \(A(l, w, h) = 2(lw + lh + wh)\), the partial derivatives with respect to each variable are:
\[ \frac{\text{∂A}}{\text{∂l}} = 2(w + h) \]
\[ \frac{\text{∂A}}{\text{∂w}} = 2(l + h) \]
\[ \frac{\text{∂A}}{\text{∂h}} = 2(l + w) \]
This shows how each dimension (length, width, height) individually affects the surface area. By substituting specific values for the dimensions of the crate (\(l = 3 \text{ ft}, w = 4 \text{ ft}, h = 5 \text{ ft}\)), we get the specific partial derivatives needed to compute the total differential.
Surface Area Calculation
Surface area calculation involves summing up the areas of all faces of an object. For a rectangular crate, you calculate the surface area by adding the areas of each face in pairs.
In the problem, the formula used is:
\[ A = 2(lw + lh + wh) \]
Given lengths \(l = 3 \text{ ft}, w = 4 \text{ ft}, h = 5 \text{ ft}\), substituting these values gives:
\[ A = 2(3 \times 4 + 3 \times 5 + 4 \times 5) = 94 \text{ square feet} \]
This calculation is necessary since the cost to prepare wood depends on the total surface area.
In the problem, the formula used is:
\[ A = 2(lw + lh + wh) \]
Given lengths \(l = 3 \text{ ft}, w = 4 \text{ ft}, h = 5 \text{ ft}\), substituting these values gives:
\[ A = 2(3 \times 4 + 3 \times 5 + 4 \times 5) = 94 \text{ square feet} \]
This calculation is necessary since the cost to prepare wood depends on the total surface area.
Error Propagation
Error propagation in calculus shows how small errors in measurements affect a calculated result. Using partial derivatives and total differentials, you can estimate the maximum error.
In the crate problem, knowing the possible error in measurements (\(0.05 \text{ ft}\)) and the partial derivatives calculated earlier, we find the propagated error in surface area using:
\[ dA = 18 \times 0.05 + 16 \times 0.05 + 14 \times 0.05 = 2.4 \text{ square feet} \]
The final step involves translating this surface area error to a cost error:
\[ \text{Error in Cost} = 5 \times 2.4 = 12 \text{ cents per crate} \]
This approach helps to predict how measurement inaccuracies affect the total cost estimate.
In the crate problem, knowing the possible error in measurements (\(0.05 \text{ ft}\)) and the partial derivatives calculated earlier, we find the propagated error in surface area using:
\[ dA = 18 \times 0.05 + 16 \times 0.05 + 14 \times 0.05 = 2.4 \text{ square feet} \]
The final step involves translating this surface area error to a cost error:
\[ \text{Error in Cost} = 5 \times 2.4 = 12 \text{ cents per crate} \]
This approach helps to predict how measurement inaccuracies affect the total cost estimate.
Other exercises in this chapter
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