https://file.aiquickdraw.com/mathbot/file/1725928393871dx8xq608.jpg

Let's solve the problem step by step:

### (a) Write down the area of the rectangle in the form \(a \times 10^k\), where \(1 \leq a < 10\) and \(k \in \mathbb{Z}\).

The dimensions of the rectangle are:
- Length: \(7.6 \times 10^2\) cm
- Width: \(1.5 \times 10^3\) cm

To find the area, we multiply the length by the width:

\[
\text{Area} = (7.6 \times 10^2) \times (1.5 \times 10^3)
\]

We can break this into two parts:
\[
= (7.6 \times 1.5) \times (10^2 \times 10^3)
\]
\[
= 11.4 \times 10^5 \ \text{cm}^2
\]

Now, express \(11.4 \times 10^5\) in scientific notation, where \(1 \leq a < 10\):

\[
= 1.14 \times 10^6 \ \text{cm}^2
\]

### (b) Find the percentage error in Natalie's estimate.

Natalie estimates the area of the rectangle to be \(1,200,000 \, \text{cm}^2\), which is \(1.2 \times 10^6\) cm².

Now, we calculate the percentage error using the formula:

\[
\text{Percentage Error} = \left( \frac{\text{Estimated Value} - \text{Actual Value}}{\text{Actual Value}} \right) \times 100
\]

Substitute the values:

\[
\text{Percentage Error} = \left( \frac{1.2 \times 10^6 - 1.14 \times 10^6}{1.14 \times 10^6} \right) \times 100
\]
\[
= \left( \frac{0.06 \times 10^6}{1.14 \times 10^6} \right) \times 100
\]
\[
= \frac{0.06}{1.14} \times 100 \approx 5.26\%
\]

Thus, the percentage error in Natalie's estimate is approximately **5.26%**.

---

Would you like more details or have any questions? Here are some related questions to consider:

1. How do you multiply numbers in scientific notation?
2. Why is it important to express answers in scientific notation?
3. Can the percentage error formula be used for other types of problems?
4. How does one determine if an estimate is reasonably accurate?
5. What are other methods to reduce error in estimates?

**Tip:** Always make sure to express your final answer in the simplest form, especially when working with scientific notation.

The following diagram shows a rectangle with sides of length 7.6 × 10² cm and 1.5 × 10³ cm. (a) Write down the area of the rectangle in the form a × 10^k, where 1 ≤ a < 10 and k ∈ Z. (b) Find the percentage error in Natalie's estimate, where she estimates the area to be 1,200,000 cm².

Learn how to calculate the area of a rectangle using scientific notation and find the percentage error in an estimate. This problem involves dimensions of 7.6 × 10² cm and 1.5 × 10³ cm. Suitable for students in Grades 9-10.

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