Đáp án:

\(f = 28,8cm\)

Giải thích các bước giải:

* Ban đầu, ta có:

\(\dfrac{{{A_1}{B_1}}}{{AB}} = \dfrac{{{d_1}'}}{{{d_1}}} = \dfrac{{\dfrac{{{d_1}f}}{{{d_1} - f}}}}{{{d_1}}} = \dfrac{f}{{{d_1} - f}}\)

* Sau khi dịch chuyển, ta có:

\(\begin{array}{l}
\dfrac{{{A_2}{B_2}}}{{AB}} = \dfrac{{{d_2}'}}{{{d_2}}} = \dfrac{{\dfrac{{{d_2}f}}{{{d_2} - f}}}}{{{d_2}}} = \dfrac{f}{{{d_2} - f}}\\
 \Rightarrow \dfrac{{{A_1}{B_1}}}{{{A_2}{B_2}}} = \dfrac{{{d_2} - f}}{{{d_1} - f}}\\
 \Rightarrow \dfrac{4}{9} = \dfrac{{{d_1} + 24 - f}}{{{d_1} - f}}\\
 \Rightarrow 4\left( {{d_1} - f} \right) = 9\left( {{d_1} + 24 - f} \right)\\
 \Rightarrow  - 5{d_1} + 5f = 216\\
 \Rightarrow f - {d_1} = 43,2
\end{array}\)

Ta có:

\(\begin{array}{l}
{d_1} + {d_1}' = {d_2} + {d_2}'\\
 \Rightarrow {d_1} + \dfrac{{{d_1}f}}{{{d_1} - f}} = {d_2} + \dfrac{{{d_2}f}}{{{d_2} - f}}\\
 \Rightarrow \dfrac{{d_1^2}}{{{d_1} - f}} = \dfrac{{d_2^2}}{{{d_2} - f}}\\
 \Rightarrow \dfrac{{{{\left( {f - 43,2} \right)}^2}}}{{f - 43,2 - f}} = \dfrac{{{{\left( {f - 43,2 + 24} \right)}^2}}}{{f - 43,2 + 24 - f}}\\
 \Rightarrow \dfrac{{{{\left( {f - 43,2} \right)}^2}}}{{ - 43,2}} = \dfrac{{{{\left( {f - 19,2} \right)}^2}}}{{ - 19,2}}\\
 \Rightarrow f = 28,8cm
\end{array}\)