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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

1 answer

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