Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 657

\[\boxed{\mathbf{657}.}\]

\[S = \sqrt{(l + r)^{2} - h^{2}} -\]

\[- \sqrt{(l - r)^{2} - h^{2}}\]

\[45 = \sqrt{(70 + 20)^{2} - h^{2}} -\]

\[- \sqrt{(70 - 20)^{2} - h^{2}}\]

\[\sqrt{90^{2} - h^{2}} - \sqrt{50^{2} - h^{2}} = 45\]

\[90^{2} - h^{2} -\]

\[- 2 \cdot \sqrt{\left( 90^{2} - h^{2} \right)\left( 50^{2} - h^{2} \right)} +\]

\[+ 50^{2} - h^{2} = 45^{2}\]

\[2 \cdot \sqrt{\left( 90^{2} - h^{2} \right)\left( 50^{2} - h^{2} \right)} =\]

\[= 8575 - 2h^{2}\]

\[4 \cdot \left( 20\ 250\ 000 - 10\ 600h^{2} + h^{4} \right) =\]

\[= 73\ 530\ 625 - 34\ 300h^{2} + 4h^{4}\]

\[2025h^{2} = 1867343,75\]

\[h = \sqrt{\frac{1867343,75}{2025}} \approx\]

\[\approx 30,366\ см \approx 3037\ мм.\]

\[Ответ:3037\ мм.\]