To solve this problem, we start by using the relationship between the length and the width of the rectangular pan. Let's denote the width of the rectangular pan by [tex]\( w \)[/tex] inches.
Given:
1. The area of the pan is 432 square inches.
2. The length [tex]\( l \)[/tex] is [tex]\(\frac{4}{3}\)[/tex] times the width [tex]\( w \)[/tex].
The formula for the area [tex]\( A \)[/tex] of a rectangle is:
[tex]\[ A = l \times w \][/tex]
Substitute the given length [tex]\( l \)[/tex] into the equation:
[tex]\[ l = \frac{4}{3}w \][/tex]
Therefore, the equation for the area becomes:
[tex]\[ 432 = \left(\frac{4}{3}w\right) \times w \][/tex]
Simplify the equation:
[tex]\[ 432 = \frac{4}{3}w^2 \][/tex]
To isolate [tex]\( w^2 \)[/tex], multiply both sides of the equation by the reciprocal of [tex]\(\frac{4}{3}\)[/tex]:
[tex]\[ 432 \times \frac{3}{4} = w^2 \][/tex]
[tex]\[ 324 = w^2 \][/tex]
Now, to find [tex]\( w \)[/tex], take the square root of both sides:
[tex]\[ w = \sqrt{324} \][/tex]
Thus, the width [tex]\( w \)[/tex] of the cake pan is:
[tex]\[ w = 18 \text{ inches} \][/tex]
So, the width of the cake pan is 18 inches.
