The volume of a rectangular prism is $150 \mathrm{~cm}^{3}$ . Alex measures the sides to be $9.89 \mathrm{~cm}$ by $3.43 \mathrm{~cm}$ by $5.08 \mathrm{~cm}$ . In calculating the volume, what is the relative error, to the nearest hundredth. $\newline$ Answer:

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Find the rate of change of one variable when rate of change of other variable is given

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Q. The volume of a rectangular prism is

150 \mathrm{~cm}^{3}

. Alex measures the sides to be

9.89 \mathrm{~cm}

3.43 \mathrm{~cm}

5.08 \mathrm{~cm}

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

\newline

Answer:

Calculate Volume: Given the volume of the rectangular prism is $150\,\text{cm}^3$ , and Alex measures the sides to be $9.89\,\text{cm}$ , $3.43\,\text{cm}$ , and $5.08\,\text{cm}$ . First, we need to calculate the volume using Alex's measurements. $\newline$ Volume = length $\times$ width $\times$ height
Find Absolute Error: Using Alex's measurements, we calculate the volume as follows: $\newline$ Volume = $9.89 \, \text{cm} \times 3.43 \, \text{cm} \times 5.08 \, \text{cm}$
Calculate Relative Error: Perform the multiplication to find the calculated volume: $\newline$ Calculated Volume = $9.89 \times 3.43 \times 5.08$ $\newline$ Calculated Volume = $171.892524 \, \text{cm}^3$
Round Relative Error: Now, we need to find the absolute error, which is the difference between the true volume and the calculated volume. $\newline$ Absolute Error = $|\text{True Volume} - \text{Calculated Volume}|$
Round Relative Error: Now, we need to find the absolute error, which is the difference between the true volume and the calculated volume. $\newline$ Absolute Error = $|\text{True Volume} - \text{Calculated Volume}|$ Substitute the given true volume and the calculated volume into the absolute error formula: $\newline$ Absolute Error = $|150 \text{ cm}^3 - 171.892524 \text{ cm}^3|$ $\newline$ Absolute Error = $| -21.892524 \text{ cm}^3|$ $\newline$ Absolute Error = $21.892524 \text{ cm}^3$
Round Relative Error: Now, we need to find the absolute error, which is the difference between the true volume and the calculated volume. $\newline$ Absolute Error = $|\text{True Volume} - \text{Calculated Volume}|$ Substitute the given true volume and the calculated volume into the absolute error formula: $\newline$ Absolute Error = $|150 \text{ cm}^3 - 171.892524 \text{ cm}^3|$ $\newline$ Absolute Error = $| -21.892524 \text{ cm}^3|$ $\newline$ Absolute Error = $21$ . $892524$ \text{ cm}^ $3$ Next, we find the relative error by dividing the absolute error by the true volume. $\newline$ Relative Error = \frac{\text{Absolute Error}}{\text{True Volume}}
Round Relative Error: Now, we need to find the absolute error, which is the difference between the true volume and the calculated volume. $\newline$ Absolute Error = $|\text{True Volume} - \text{Calculated Volume}|$ Substitute the given true volume and the calculated volume into the absolute error formula: $\newline$ Absolute Error = $|150 \text{ cm}^3 - 171.892524 \text{ cm}^3|$ $\newline$ Absolute Error = $| -21.892524 \text{ cm}^3|$ $\newline$ Absolute Error = $21$ . $892524$ \text{ cm}^ $3$ Next, we find the relative error by dividing the absolute error by the true volume. $\newline$ Relative Error = \frac{\text{Absolute Error}}{\text{True Volume}}Substitute the absolute error and the true volume into the relative error formula: $\newline$ Relative Error = \frac{ $21$ . $892524$ \text{ cm}^ $3$ }{ $150$ \text{ cm}^ $3$ }
Round Relative Error: Now, we need to find the absolute error, which is the difference between the true volume and the calculated volume. $\newline$ Absolute Error = $|\text{True Volume} - \text{Calculated Volume}|$ Substitute the given true volume and the calculated volume into the absolute error formula: $\newline$ Absolute Error = $|150 \text{ cm}^3 - 171.892524 \text{ cm}^3|$ $\newline$ Absolute Error = $| -21.892524 \text{ cm}^3|$ $\newline$ Absolute Error = $21$ . $892524$ \text{ cm}^ $3$ Next, we find the relative error by dividing the absolute error by the true volume. $\newline$ Relative Error = \frac{\text{Absolute Error}}{\text{True Volume}}Substitute the absolute error and the true volume into the relative error formula: $\newline$ Relative Error = \frac{ $21$ . $892524$ \text{ cm}^ $3$ }{ $150$ \text{ cm}^ $3$ }Perform the division to find the relative error: $\newline$ Relative Error = $0$ . $14595016$
Round Relative Error: Now, we need to find the absolute error, which is the difference between the true volume and the calculated volume. $\newline$ Absolute Error = $|\text{True Volume} - \text{Calculated Volume}|$ Substitute the given true volume and the calculated volume into the absolute error formula: $\newline$ Absolute Error = $|150 \text{ cm}^3 - 171.892524 \text{ cm}^3|$ $\newline$ Absolute Error = $| -21.892524 \text{ cm}^3|$ $\newline$ Absolute Error = $21$ . $892524$ \text{ cm}^ $3$ Next, we find the relative error by dividing the absolute error by the true volume. $\newline$ Relative Error = \frac{\text{Absolute Error}}{\text{True Volume}}Substitute the absolute error and the true volume into the relative error formula: $\newline$ Relative Error = \frac{ $21$ . $892524$ \text{ cm}^ $3$ }{ $150$ \text{ cm}^ $3$ }Perform the division to find the relative error: $\newline$ Relative Error = $0$ . $14595016$ To express the relative error to the nearest hundredth, we round the result to two decimal places: $\newline$ Relative Error \approx $0$ . $15$