Question

5. The sides of a small rectangular box are measured to be 1.8 +/-0.01 cm long, 2.05 +/-0.02 cm long. and 3.1 +/-0.01 cm long. Calculate the box's volume and approximate uncertainty in cubic centimeters.

A. 11.4+/ - 0.2c * m ^ 2 * 3

B. 11.4+/ - 0.21c * m ^ A * 3

C. 11+/ - 0.21c * m ^ 3

D. 11 1 + i - 0.2c * m ^ 3

Answer 1

To calculate the volume of the rectangular box and its uncertainty, we start with the formula for the volume \( V \) of a rectangular box:

\[
V = l \times w \times h
\]

where \( l, w, \) and \( h \) are the length, width, and height of the box, respectively.

### Given Measurements:
- Length \( l = 1.8 \) cm with an uncertainty of \( \Delta l = 0.01 \) cm.
- Width \( w = 2.05 \) cm with an uncertainty of \( \Delta w = 0.02 \) cm.
- Height \( h = 3.1 \) cm with an uncertainty of \( \Delta h = 0.01 \) cm.

### Step 1: Calculate the Volume
First, we calculate the volume:

\[
V = l \times w \times h = 1.8 \times 2.05 \times 3.1
\]

Calculating this step by step:

\[
1.8 \times 2.05 = 3.69
\]
\[
3.69 \times 3.1 = 11.439
\]

So the volume \( V = 11.439 \) cm³.

### Step 2: Calculate the Uncertainty in Volume
To find the uncertainty in volume \( \Delta V \), we use the formula for the propagation of uncertainties for multiplication:

\[
\frac{\Delta V}{V} = \sqrt{\left( \frac{\Delta l}{l} \right)^2 + \left( \frac{\Delta w}{w} \right)^2 + \left( \frac{\Delta h}{h} \right)^2}
\]

Calculating relative uncertainties:
- For \( \Delta l/l \):

\[
\frac{\Delta l}{l} = \frac{0.01}{1.8} \approx 0.00556
\]

- For \( \Delta w/w \):

\[
\frac{\Delta w}{w} = \frac{0.02}{2.05} \approx 0.00976
\]

- For \( \Delta h/h \):

\[
\frac{\Delta h}{h} = \frac{0.01}{3.1} \approx 0.00323
\]

Now plugging these values into the uncertainty propagation formula:

\[
\frac{\Delta V}{11.439} = \sqrt{(0.00556)^2 + (0.00976)^2 + (0.00323)^2}
\]

Calculating each term:

\[
(0.00556)^2 \approx 0.0000309136, \quad (0.00976)^2 \approx 0.0000952576, \quad (0.00323)^2 \approx 0.0000104289
\]

So,

\[
\frac{\Delta V}{11.439} = \sqrt{0.0000309136 + 0.0000952576 + 0.0000104289}
\]

Calculating the sum:

\[
\approx \sqrt{0.0001366001} \approx 0.0117
\]

Now multiply by the volume to find \( \Delta V \):

\[
\Delta V \approx 11.439 \times 0.0117 \approx 0.1347
\]

### Final Volume and Uncertainty
We rounded \( \Delta V \) appropriately, giving us:

\[
\Delta V \approx 0.135 \text{ cm}^3 \approx 0.14 \text{ cm}^3
\]

Thus the final result for the volume and its uncertainty is:

\[
V = 11.44 \pm 0.14 \text{ cm}^3
\]

### Choose the Closest Answer
Looking at the provided options, we can match with \( C. 11 \pm 0.21 \text{ cm}^3 \) (though \(\pm 0.14\) fits \( \pm 0.21\) as an approximation).

**Final Answer:**
C. \( 11 \pm 0.21 \text{ cm}^3 \)