Question

c) if the width or expression for the length of rectangle b?

1. a rectangular wall has a circular window. the area of the wall can be represented by ( 32x^2 + 16x ). the length of the wall is ( 8x ). the diameter of the circular window has a measurement that is half the width of the wall. what is the radius of the window written as an expression in terms of ( x )?

( 8x )
( a = 32x^2 + 16x )

Explanation:

Step1: Find the width of the wall

The area of a rectangle is given by \( A = \text{length} \times \text{width} \). We know \( A = 32x^{2}+16x \) and length \( = 8x \). So, width \( w=\frac{A}{\text{length}}=\frac{32x^{2} + 16x}{8x} \).
Simplify the fraction: \( \frac{32x^{2}}{8x}+\frac{16x}{8x}=4x + 2 \).

Step2: Find the diameter of the window

The diameter of the window is half the width of the wall. So, diameter \( d=\frac{1}{2}(4x + 2) \).
Simplify: \( d = 2x+1 \).

Step3: Find the radius of the window

The radius \( r \) of a circle is half of its diameter. So, \( r=\frac{d}{2}=\frac{2x + 1}{2}=x+\frac{1}{2} \).

Answer:

The radius of the window is \( x+\frac{1}{2} \) (or \( \frac{2x + 1}{2} \)).