To calculate the uncertainty in the area of the garden, given the lengths with their uncertainties, we follow this process:

Given:

• Length ($L$) = 4.40 ± 0.46 m
• Width ($W$) = 5.75 ± 0.44 m
• Formula for area $A = L \times W$

Step 1: Calculate the area $A$

$A = L \times W = 4.40 \times 5.75 = 25.3 \, \text{m}^2$

Step 2: Determine the fractional uncertainties

The fractional uncertainty for each dimension is calculated as the uncertainty divided by the value:

$\frac{\Delta L}{L} = \frac{0.46}{4.40} = 0.1045$ $\frac{\Delta W}{W} = \frac{0.44}{5.75} = 0.0765$

Step 3: Calculate the total fractional uncertainty for the area

For multiplication, the total fractional uncertainty is the sum of the fractional uncertainties of length and width:

$\frac{\Delta A}{A} = \frac{\Delta L}{L} + \frac{\Delta W}{W} = 0.1045 + 0.0765 = 0.181$

Step 4: Calculate the absolute uncertainty in the area

Now, multiply the total fractional uncertainty by the area to get the absolute uncertainty:

$\Delta A = A \times \frac{\Delta A}{A} = 25.3 \times 0.181 = 4.58 \, \text{m}^2$

Final Answer:

The uncertainty in the area is 4.6 m² (rounded to two significant figures).

Would you like further clarification on any part of the calculation?

Here are 5 related questions you might find useful:

1. How do you calculate uncertainty when adding or subtracting quantities?
2. What is the difference between absolute and fractional uncertainty?
3. How do uncertainties combine when multiplying or dividing more than two