The volume of a rectangular prism is $54 \mathrm{ft}^{3}$ . Carter measures the sides to be $2.85 \mathrm{ft}$ by $9.32 \mathrm{ft}$ by $2.09 \mathrm{ft}$ . In calculating the volume, what is the relative error, to the nearest thousandth. $\newline$ Answer:

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Q. The volume of a rectangular prism is

54 \mathrm{ft}^{3}

. Carter measures the sides to be

2.85 \mathrm{ft}

9.32 \mathrm{ft}

2.09 \mathrm{ft}

. In calculating the volume, what is the relative error, to the nearest thousandth.

\newline

Answer:

Given: Given: $\newline$ Actual volume $V_{\text{actual}}$ = $54\,\text{ft}^3$ $\newline$ Measured sides are $2.85\,\text{ft}$ , $9.32\,\text{ft}$ , and $2.09\,\text{ft}$ . $\newline$ To find the calculated volume $V_{\text{calculated}}$ , we multiply the measured sides: $\newline$ $V_{\text{calculated}} = 2.85\,\text{ft} \times 9.32\,\text{ft} \times 2.09\,\text{ft}$
Perform multiplication: Perform the multiplication to find the calculated volume: $\newline$ $V_{\text{calculated}} = 2.85 \times 9.32 \times 2.09$ $\newline$ $V_{\text{calculated}} = 55.59666\text{ft}^3$ (rounded to five decimal places for intermediate calculation)
Calculate relative error: The relative error is calculated using the formula: $\newline$ Relative Error = $\left| V_{\text{actual}} - V_{\text{calculated}} \right| / V_{\text{actual}}$ $\newline$ First, find the absolute difference between the actual volume and the calculated volume: $\newline$ Difference = $\left| V_{\text{actual}} - V_{\text{calculated}} \right|$ $\newline$ Difference = $\left| 54 - 55.59666 \right|$ $\newline$ Difference = $1.59666$ ft³ (rounded to five decimal places for intermediate calculation)
Find absolute difference: Now, calculate the relative error using the difference found: $\newline$ Relative Error = $\frac{\text{Difference}}{V_{\text{actual}}}$ $\newline$ Relative Error = $\frac{1.59666}{54}$ $\newline$ Relative Error $\approx 0.029568$ (rounded to six decimal places for intermediate calculation)
Calculate relative error: Round the relative error to the nearest thousandth as requested: $\newline$ Relative Error $\approx 0.030$ (to the nearest thousandth)