To solve for the width of the rectangular pan, let's proceed with a step-by-step approach.
1. Express the Relationship Between Length and Width:
We are told the length (L) of the pan is [tex]\(\frac{4}{3}\)[/tex] times the width (W).
[tex]\[ L = \frac{4}{3}W \][/tex]
2. Express the Area in Terms of Width:
The area (A) of a rectangle is given by length times width.
[tex]\[ A = L \times W \][/tex]
Substituting the expression for length ([tex]\(L = \frac{4}{3}W\)[/tex]):
[tex]\[ A = \left(\frac{4}{3}W\right) \times W \][/tex]
Simplifying the right-hand side:
[tex]\[ A = \frac{4}{3}W^2 \][/tex]
3. Substitute the Given Area:
We are given that the total area of the pan is [tex]\(432 \, \text{in}^2\)[/tex].
[tex]\[ 432 = \frac{4}{3}W^2 \][/tex]
4. Solve for the Width (W):
[tex]\[ 432 = \frac{4}{3}W^2 \][/tex]
To isolate [tex]\(W^2\)[/tex], multiply both sides by [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[ 432 \times \frac{3}{4} = W^2 \][/tex]
[tex]\[ 324 = W^2 \][/tex]
To find [tex]\(W\)[/tex], take the square root of both sides:
[tex]\[ W = \sqrt{324} \][/tex]
[tex]\[ W = 18 \, \text{in} \][/tex]
The width of the cake pan is [tex]\( \boxed{18 \, \text{in}} \)[/tex].
