Question

A rectangle has length 9.3 × 10^2 cm and width 8.1×10 cm. Which of these statements are true?(4 points)
True | False
The area of the rectangle is 7.533 × 10^4 cm^2:


The perimeter of the rectangle is 1.71 × 10^3 cm:


The length of the rectangle is approximately 1.15 × 10 times greater than the width:


The length of the rectangle squared is approximately 8.65 × 10^5

Answer 1

Let's evaluate each of the statements one by one.

1. **Area of the rectangle**:
The area \( A \) of a rectangle is given by the formula:
\[
A = \text{length} \times \text{width}
\]
Given length \( = 9.3 \times 10^2 \) cm and width \( = 8.1 \times 10^1 \) cm, we calculate the area:
\[
A = (9.3 \times 10^2) \times (8.1 \times 10^1) = 9.3 \times 8.1 \times 10^{2 + 1}
\]
Calculating \( 9.3 \times 8.1 = 75.33 \):
\[
A = 75.33 \times 10^3 = 7.533 \times 10^4 \text{ cm}^2
\]
The first statement is **True**.

2. **Perimeter of the rectangle**:
The perimeter \( P \) of a rectangle is given by the formula:
\[
P = 2 \times (\text{length} + \text{width})
\]
Using the same values:
\[
P = 2 \times ((9.3 \times 10^2) + (8.1 \times 10^1))
\]
We first calculate:
\[
8.1 \times 10^1 = 81 \text{ cm} \quad \text{then} \quad (\text{length} + \text{width}) = 930 + 81 = 1011
\]
So:
\[
P = 2 \times 1011 = 2022 \text{ cm} = 2.022 \times 10^3 \text{ cm}
\]
The perimeter statement \( 1.71 \times 10^3 \) cm is incorrect. The second statement is **False**.

3. **Length compared to width**:
The length of the rectangle is \( 9.3 \times 10^2 \), and the width is \( 8.1 \times 10^1 \). To compare:
\[
\frac{\text{length}}{\text{width}} = \frac{9.3 \times 10^2}{8.1 \times 10^1} = \frac{9.3}{8.1} \times 10^{2 - 1} = \frac{9.3}{8.1} \times 10^{1}
\]
Calculating \( \frac{9.3}{8.1} \approx 1.148 \):
\[
\text{Thus,} \quad \frac{9.3}{8.1} \approx 1.15 \quad \text{(approximately)}
\]
This means the length is approximately \( 1.15 \times 10^1 = 1.15 \times 10 \) times greater than the width. This statement is **True**.

4. **Length squared**:
We need to find the square of the length:
\[
\text{Length}^2 = (9.3 \times 10^2)^2 = 9.3^2 \times (10^2)^2 = 86.49 \times 10^4
\]
To express this in scientific notation:
\[
86.49 \times 10^4 = 8.649 \times 10^5
\]
The statement claims the length squared is approximately \( 8.65 \times 10^5 \). Given that \( 8.649 \) is indeed very close to \( 8.65 \), we can accept this as **True**.

In summary:
- Area: **True**
- Perimeter: **False**
- Length compared to width: **True**
- Length squared: **True**