19.) Error Analysis A student calculates the area and perimeter of a rectangle with \[ \begin{array}{l} \text { length } 7.5 \mathrm{~cm} \text { and width } 3 \mathrm{~cm} \text { as follows: } \\ \)\[ \]\( =3(7.5) \quad \text { and } \quad P=3+7.5 \\ \text { Explain the student's error. }\end{array} \]

The student made errors in both the calculation of the area and the perimeter of the rectangle. 1. **Area Calculation Error**: - The correct formula for the area \( A \) of a rectangle is given by: \[ A = \text{length} \times \text{width} \] - For the given rectangle, the length is \( 7.5 \, \text{cm} \) and the width is \( 3 \, \text{cm} \). Therefore, the correct calculation should be: \[ A = 7.5 \times 3 = 22.5 \, \text{cm}^2 \] - The student incorrectly wrote \( A = 3(7.5) \), which is not the correct representation of the area formula. While this expression does yield the correct numerical result (since \( 3 \times 7.5 = 22.5 \)), it is misleading because it does not clearly indicate that the area is calculated by multiplying length and width. 2. **Perimeter Calculation Error**: - The correct formula for the perimeter \( P \) of a rectangle is given by: \[ P = 2 \times (\text{length} + \text{width}) \] - For the given rectangle, the correct calculation should be: \[ P = 2 \times (7.5 + 3) = 2 \times 10.5 = 21 \, \text{cm} \] - The student incorrectly wrote \( P = 3 + 7.5 \), which only adds the length and width together without multiplying by 2. This results in an incorrect perimeter of \( 10.5 \, \text{cm} \) instead of the correct \( 21 \, \text{cm} \). In summary, the student's errors stem from misunderstanding the formulas for area and perimeter, leading to incorrect calculations. The area should be calculated by multiplying length and width, and the perimeter should be calculated by adding the length and width together and then multiplying by 2.