https://file.aiquickdraw.com/mathbot/file/1733933763954e24mzlsv.jpg

### Problem Analysis:
The task involves determining the **upper bound for the perimeter** and the **lower bound for the area** of a rectangle, given measurements that are rounded to the nearest tenth of a meter.

#### Dimensions:
- **Length:** 15.8 m (rounded to nearest tenth)
- **Width:** 5.2 m (rounded to nearest tenth)

When measurements are rounded to the nearest tenth:
- **Upper Bound:** Add 0.05 to the rounded measurement.
- **Lower Bound:** Subtract 0.05 from the rounded measurement.

### Step-by-Step Solution:

#### a) **Upper Bound for the Perimeter:**
The perimeter \( P \) of a rectangle is given by:
\[
P = 2(\text{Length} + \text{Width})
\]

- **Upper bound of length:** \( 15.8 + 0.05 = 15.85 \, \text{m} \)
- **Upper bound of width:** \( 5.2 + 0.05 = 5.25 \, \text{m} \)

Substitute the upper bounds into the formula:
\[
P_{\text{upper}} = 2(15.85 + 5.25) = 2(21.10) = 42.2 \, \text{m}
\]

#### b) **Lower Bound for the Area:**
The area \( A \) of a rectangle is given by:
\[
A = \text{Length} \times \text{Width}
\]

- **Lower bound of length:** \( 15.8 - 0.05 = 15.75 \, \text{m} \)
- **Lower bound of width:** \( 5.2 - 0.05 = 5.15 \, \text{m} \)

Substitute the lower bounds into the formula:
\[
A_{\text{lower}} = 15.75 \times 5.15 = 81.1125 \, \text{m}^2
\]

### Final Answers:
1. **Upper bound for the perimeter:** \( 42.2 \, \text{m} \)
2. **Lower bound for the area:** \( 81.1125 \, \text{m}^2 \)

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Let me know if you would like further clarification or details about the calculations.

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### Related Questions:
1. What is the lower bound for the perimeter of the rectangle?
2. What is the upper bound for the area of the rectangle?
3. How would the calculations change if measurements were rounded to the nearest hundredth?
4. Why do upper and lower bounds matter in real-world measurements?
5. How can rounding errors affect calculations in construction or engineering?

### Tip:
Always double-check units and rounding increments when calculating bounds to ensure accuracy.

The lengths of the rectangle have been measured to the nearest tenth of a meter. Work out the following, writing down all the figures on your calculator display:
a) The upper bound for the perimeter of the rectangle.
b) The lower bound for the area of the rectangle.

Dimensions:
- Length: 15.8 m
- Width: 5.2 m

Learn how to calculate the upper bound for the perimeter and the lower bound for the area of a rectangle with lengths measured to the nearest tenth of a meter. Includes key mathematical concepts, step-by-step solutions, and formulas for Grade 8-10 students.

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