Questions: Part c (1 points) As a final surprise celebration, Riley decides to fill the room with balloons, and therefore calculates the volume of the room as V=W × H × D. Which of the following expressions properly describes the uncertainty in Riley's calculated volume? Select the correct answer (delta V/V)=sqrt((delta W/W)^2+(delta D/D)^2+(delta H/H)^2) (delta V/V)=(delta W/W)+(delta D/D)+(delta H/H) (delta V/V)=sqrt((delta W/W)^2+(delta D/D)^2)+(delta H/H) delta V=(delta W+delta D) × delta H (Delta V/V)=(delta W/W) × (delta D/D) × (delta H/H)

Part c
(1 points)
As a final surprise celebration, Riley decides to fill the room with balloons, and therefore calculates the volume of the room as V=W × H × D. Which of the following expressions properly describes the uncertainty in Riley's calculated volume?

Select the correct answer
(delta V/V)=sqrt((delta W/W)^2+(delta D/D)^2+(delta H/H)^2)
(delta V/V)=(delta W/W)+(delta D/D)+(delta H/H)
(delta V/V)=sqrt((delta W/W)^2+(delta D/D)^2)+(delta H/H)
delta V=(delta W+delta D) × delta H
(Delta V/V)=(delta W/W) × (delta D/D) × (delta H/H)

Transcript text: Part c (1 points) As a final surprise celebration, Riley decides to fill the room with balloons, and therefore calculates the volume of the room as $V=W \times H \times D$. Which of the following expressions properly describes the uncertainty in Riley's calculated volume? Select the correct answer $\frac{\delta V}{V}=\sqrt{\left(\frac{\delta W}{W}\right)^{2}+\left(\frac{\delta D}{D}\right)^{2}+\left(\frac{\delta H}{H}\right)^{2}}$ $\frac{\delta V}{V}=\frac{\delta W}{W}+\frac{\delta D}{D}+\frac{\text { delta } H}{H}$ $\frac{\delta V}{V}=\sqrt{\left(\frac{\delta W}{W}\right)^{2}+\left(\frac{\delta D}{D}\right)^{2}}+\frac{\delta H}{H}$ $\delta V=(\delta W+\delta D) \times \delta H$ $\frac{\Delta V}{V}=\frac{\delta W}{W} \times \frac{\delta D}{D} \times \frac{\text { delta } H}{H}$

Solution

Solution Steps

Step 1: Understand the Problem

The problem asks us to determine the correct expression for the uncertainty in the calculated volume $ V $ of a room, given by the formula $ V = W \times H \times D $. The uncertainties in the measurements of width $ W $, height $ H $, and depth $ D $ are given as $ \delta W $, $ \delta H $, and $ \delta D $, respectively.

Step 2: Apply the Rule for Propagation of Uncertainty

When dealing with the multiplication of independent variables, the relative uncertainty in the product is the square root of the sum of the squares of the relative uncertainties of the individual factors. This is a standard rule for the propagation of uncertainty in products.

Step 3: Identify the Correct Expression

Given the rule for propagation of uncertainty, the correct expression for the relative uncertainty in the volume $ V $ is:

\[ \frac{\delta V}{V} = \sqrt{\left(\frac{\delta W}{W}\right)^{2} + \left(\frac{\delta D}{D}\right)^{2} + \left(\frac{\delta H}{H}\right)^{2}} \]

This expression accounts for the uncertainties in all three dimensions (width, height, and depth) and is consistent with the rule for propagation of uncertainty in products.

Final Answer

\[ \boxed{\frac{\delta V}{V} = \sqrt{\left(\frac{\delta W}{W}\right)^{2} + \left(\frac{\delta D}{D}\right)^{2} + \left(\frac{\delta H}{H}\right)^{2}}} \]

Was this solution helpful?

Unhelpful

Helpful

Questions asked by other users

Next >

Thank you for your feedback.

Copied!