Problem 19
Question
{Volume and Surface Area } The radius of a sphere is measured to be 6 inches, with a possible error of 0.02 inch. Use differentials to approximate the maximum possible error in calculating (a) the volume of the sphere, (b) the surface area of the sphere, and (c) the relative errors in parts (a) and (b).
Step-by-Step Solution
Verified Answer
The maximum possible errors in calculating the volume and the surface area are \( dV \) and \( dA \) respectively. The relative errors for these calculations are \( \frac{dV}{V} \) and \( \frac{dA}{A} \) respectively.
1Step 1: Finding the Volume Differential
First, the formula for the volume of the sphere is given by \( V = \frac{4}{3} \pi r^{3} \) where \( r \) is the radius. By differentiating this function with respect to \( r \), we have \( dV = 4\pi r^{2} dr \). We are given \( dr = 0.02 \) inches as the possible error in the radius, so substituting this and the given value of \( r = 6 \) inches, the maximum possible error in the volume calculation, \( dV \), can be found.
2Step 2: Finding the Surface Area Differential
Next, using the formula for the surface area of the sphere \( A = 4\pi r^{2} \), differentiating with respect to \( r \) gives \( dA = 8\pi r dr \). Substituting the given values \( r = 6 \) inches and \( dr = 0.02 \) inches, we find the maximum possible error in the surface area calculation, \( dA \).
3Step 3: Calculating the Relative Errors
The relative error is just the ratio of the absolute error to the measured value. In this context it's the ratio of the differential of the function to the value of the function. Thereby, the relative error in the volume is \( \frac{dV}{V} \) and the relative error in the surface area is \( \frac{dA}{A} \). We calculate these values using the previously computed values of \( dV \) and \( dA \), and the formulas for \( V \) and \( A \) at \( r = 6 \) inches.
Key Concepts
Volume of a SphereSurface Area of a SphereRelative Error in CalculationsError Propagation
Volume of a Sphere
The concept of the volume of a sphere is fundamental in geometry. The formula for calculating the volume is given by \[ V = \frac{4}{3} \pi r^3 \]where \( V \) represents the volume and \( r \) is the radius. This formula shows that the volume is directly related to the cube of the radius.
When measuring the volume of a sphere, any error in the radius will significantly affect the final volume because changing the radius slightly will alter its cube. If the radius is measured with a small error like \( dr \), it can lead to maximum possible error in the volume, calculated using differentials.
Differentials provide an approximation of how a small change in one variable can affect changes in the function's output. For the sphere, the differential \( dV \) is determined by differentiating the volume formula with respect to \( r \), leading to the formula \[ dV = 4\pi r^2 dr \].Small changes in \( r \) are magnified when cubed, thus significantly impacting the volume.
When measuring the volume of a sphere, any error in the radius will significantly affect the final volume because changing the radius slightly will alter its cube. If the radius is measured with a small error like \( dr \), it can lead to maximum possible error in the volume, calculated using differentials.
Differentials provide an approximation of how a small change in one variable can affect changes in the function's output. For the sphere, the differential \( dV \) is determined by differentiating the volume formula with respect to \( r \), leading to the formula \[ dV = 4\pi r^2 dr \].Small changes in \( r \) are magnified when cubed, thus significantly impacting the volume.
Surface Area of a Sphere
Surface area defines the measure of the total area that the surface of the sphere occupies. The formula to find the surface area of a sphere is given by \[ A = 4\pi r^2 \].This equation shows the dependence of surface area on the square of the radius \( r \).
Any error in measuring the radius affects the surface area of the sphere. To determine the maximum possible error in calculating the surface area due to error \( dr \) in the radius, we use the differential formula \[ dA = 8\pi r dr \].This helps to approximate how a small error \( dr \) can lead to an error in the surface area.
Differentiating the surface area formula allows us to estimate the error propagation from the radial measurement to the surface area. It demonstrates the sensitivity of the surface area to small errors in radius measurement.
Any error in measuring the radius affects the surface area of the sphere. To determine the maximum possible error in calculating the surface area due to error \( dr \) in the radius, we use the differential formula \[ dA = 8\pi r dr \].This helps to approximate how a small error \( dr \) can lead to an error in the surface area.
Differentiating the surface area formula allows us to estimate the error propagation from the radial measurement to the surface area. It demonstrates the sensitivity of the surface area to small errors in radius measurement.
Relative Error in Calculations
Relative error is crucial for understanding the accuracy of a measurement relative to the size of the measurement itself. It is essentially the ratio between the absolute error of a measurement and the actual measurement value.
For the volume of a sphere, the relative error is expressed as\[ \frac{dV}{V} \],where \( dV \) is the differential of volume, and \( V \) is the volume.
For the surface area of the sphere, the relative error is\[ \frac{dA}{A} \],where \( dA \) is the differential of surface area, and \( A \) is the surface area.
Calculating these relative errors gives insight into the accuracy and reliability of the measurements. It shows how significant an absolute error is in the context of the dimensions involved.
For the volume of a sphere, the relative error is expressed as\[ \frac{dV}{V} \],where \( dV \) is the differential of volume, and \( V \) is the volume.
For the surface area of the sphere, the relative error is\[ \frac{dA}{A} \],where \( dA \) is the differential of surface area, and \( A \) is the surface area.
Calculating these relative errors gives insight into the accuracy and reliability of the measurements. It shows how significant an absolute error is in the context of the dimensions involved.
Error Propagation
Error propagation refers to how uncertainties in measurements carry through calculations to affect results. In the context of a sphere, variations in the radius can lead to errors in both the volume and surface area calculations.
Differentials allow us to estimate these propagated errors by showing how small changes in the radius lead to changes in related calculations. Understanding error propagation is vital for scientists and engineers who must accurately predict how uncertainties impact results in applied mathematics and science.
Differentials allow us to estimate these propagated errors by showing how small changes in the radius lead to changes in related calculations. Understanding error propagation is vital for scientists and engineers who must accurately predict how uncertainties impact results in applied mathematics and science.
- It provides a quantitative way of predicting how error in measurements can influence outcomes.
- It is crucial for ensuring precision and reliability in scientific and industrial calculations.
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