The problem in the image seems to be asking to measure the length $a$ and width $b$ of a rectangular object and then calculate the perimeter and area.

• $a = 100 \pm 0.2 \, \text{cm}$
• $b = 50 \pm 0.5 \, \text{cm}$

It mentions calculating the perimeter and area, taking into account the uncertainties.

To solve this:

1. Perimeter (P) of a rectangle: $P = 2(a + b)$ Substituting the values: $P = 2(100 + 50) = 300 \, \text{cm}$

2. Area (A) of a rectangle: $A = a \times b$ Substituting the values: $A = 100 \times 50 = 5000 \, \text{cm}^2$

Now, the uncertainties:

• Uncertainty in Perimeter: The uncertainty in the perimeter is: $\Delta P = 2 (\Delta a + \Delta b) = 2(0.2 + 0.5) = 1.4 \, \text{cm}$

  So, the perimeter with uncertainty is: $P = 300 \pm 1.4 \, \text{cm}$

• Uncertainty in Area: The uncertainty in the area is calculated as: $\Delta A = A \times \left( \frac{\Delta a}{a} + \frac{\Delta b}{b} \right)$ Substituting the values: $\Delta A = 5000 \times \left( \frac{0.2}{100} + \frac{0.5}{50} \right) = 5000 \times 0.014 = 70 \, \text{cm}^2$ So, the area with uncertainty is: $A = 5000 \pm 70 \, \text{cm}^2$

Would you like more details or have any questions on this?

Here are 5 related questions:

1. How do uncertainties affect the accuracy of measurements?
2. What other geometric shapes can we calculate perimeter and area for?
3. How do you calculate uncertainties for more complex formulas like volumes?
4. What units are commonly used for measuring area and how do they convert?
5. Why is it important to consider uncertainty in measurements?

Tip: When multiplying or dividing measurements, always add the relative uncertainties (percentage uncertainty) to find the total uncertainty.