Question

1. a) a rectangle has an area of ( 36x^4 , \text{cm}^2 ). if the width measures ( 9x ), calculate the length.

Explanation:

Step1: Recall area formula for rectangle

The area \( A \) of a rectangle is given by \( A = \text{length} \times \text{width} \), so \( \text{length} = \frac{A}{\text{width}} \).

Step2: Substitute given values

Given \( A = 36x^4 \, \text{cm}^2 \) and width \( = 9x \, \text{cm} \), substitute into the formula: \( \text{length} = \frac{36x^4}{9x} \).

Step3: Simplify the expression

Simplify the coefficients and the variables separately. For coefficients: \( \frac{36}{9} = 4 \). For variables: using the rule \( \frac{x^m}{x^n} = x^{m - n} \), we have \( \frac{x^4}{x} = x^{4 - 1} = x^3 \). Multiply the results: \( 4 \times x^3 = 4x^3 \).

Answer:

The length of the rectangle is \( 4x^3 \) cm.