Sure, let me walk you through the steps of solving this problem.
John wants to make a rectangular mosaic with an area of 48 square centimeters. Let's denote the width of the rectangle by [tex]\( w \)[/tex] cm.
According to the problem, the length of the rectangle is 2 cm longer than the width, so we can express the length as [tex]\( w + 2 \)[/tex] cm.
The area of a rectangle is given by the product of its length and width. Therefore, the equation for the area can be written as:
[tex]\[ \text{width} \times \text{length} = \text{area} \][/tex]
Substituting the width [tex]\( w \)[/tex] and the length [tex]\( w + 2 \)[/tex] into the equation:
[tex]\[ w \times (w + 2) = 48 \][/tex]
Let's simplify and verify it step-by-step:
1. Write the area formula with given values:
[tex]\[ w \times (w + 2) = 48 \][/tex]
This equation shows the relationship between the width [tex]\( w \)[/tex] and the area of the mosaic.
So, the correct equation that John should solve to find the width [tex]\( w \)[/tex] is:
[tex]\[ w(w + 2) = 48 \][/tex]
Thus, the equation John could solve to find [tex]\( w \)[/tex], the greatest width in centimeters he can use for the mosaic, is:
[tex]\[ w(w + 2) = 48 \][/tex]
