To solve for [tex]\(\ell\)[/tex] in terms of [tex]\(A\)[/tex], [tex]\(w\)[/tex], and [tex]\(h\)[/tex] from the equation [tex]\(A = 2\ell w + 2 w h + 2 h \ell\)[/tex], we can start by simplifying and factoring the equation.
Given:
[tex]\[ A = 2\ell w + 2 w h + 2 h \ell \][/tex]
First, factor out the common factor of 2 from the terms on the right-hand side:
[tex]\[ A = 2(\ell w + w h + h \ell) \][/tex]
Next, divide both sides of the equation by 2 to further simplify it:
[tex]\[ \frac{A}{2} = \ell w + w h + h \ell \][/tex]
Rearrange the terms on the right-hand side to group all terms involving [tex]\(\ell\)[/tex] together:
[tex]\[ \frac{A}{2} = \ell w + h \ell + w h \][/tex]
Factor [tex]\(\ell\)[/tex] out of the terms it appears in:
[tex]\[ \frac{A}{2} = \ell (w + h) + w h \][/tex]
Now, isolate the term involving [tex]\(\ell\)[/tex] by subtracting [tex]\(w h\)[/tex] from both sides:
[tex]\[ \frac{A}{2} - w h = \ell (w + h) \][/tex]
Finally, solve for [tex]\(\ell\)[/tex] by dividing both sides of the equation by [tex]\((w + h)\)[/tex]:
[tex]\[ \ell = \frac{\frac{A}{2} - w h}{w + h} \][/tex]
Simplify the expression:
[tex]\[ \ell = \frac{A - 2 w h}{2(w + h)} \][/tex]
Therefore, the solution for [tex]\(\ell\)[/tex] in terms of [tex]\(A\)[/tex], [tex]\(w\)[/tex], and [tex]\(h\)[/tex] is:
[tex]\[
\ell = \frac{A - 2 w h}{2(w + h)}
\][/tex]
