15. \( \left\{\begin{array}{l}\text { EKUB } a ; b)=1 \\ \frac{a+b}{a-b}=\frac{14}{9}\end{array}\right. \) sistemasidan foydalanib, \( a \cdot b \) ning qiymatini toping. \( \begin{array}{llll}\text { A) } 5 & \text { B) } 115 & \text { C) } 23 & \text { D) } 28\end{array} \) \( \begin{array}{llll}\text { 16. Radiusi } 4 \sqrt{3} \text { bo'lgan sharga ichki } \\ \text { chizilgan silindr hajmining eng katta }\end{array} \) qiymatini toping. \( \begin{array}{llll}\text { A) } 128 \pi & \text { B) } 324 \pi & \text { C) } 256 \pi & \text { D) } 216 \pi\end{array} \)

### 15-misol Berilgan shartlar: \[ \gcd(a,b)=1,\quad \frac{a+b}{a-b}=\frac{14}{9}. \] Avvalo, hosil qilamiz: \[ \frac{a+b}{a-b}=\frac{14}{9} \implies 9(a+b)=14(a-b). \] Bu tenglamani yechamiz: \[ 9a+9b=14a-14b, \] \[ 9a+9b-14a+14b=0, \] \[ -5a+23b=0 \quad \text{yoki} \quad 5a=23b. \] Shundan: \[ a=\frac{23}{5}b. \] Chunki \(a\) va \(b\) butun sonlar bo‘lishi va \(\gcd(a,b)=1\), \(b\) \(5\) ga bo‘linishi shart. Demak, \(b=5k\) bo‘lsa: \[ a=23k. \] Ikki sonning EKUBi: \[ \gcd(23k,5k)=k\gcd(23,5)=k, \] chunki \(23\) va \(5\) bir-biriga tub. Shartga ko‘ra \(\gcd(a,b)=1\) bo‘lishi uchun \(k=1\). Natijada: \[ a=23,\quad b=5. \] So‘ng \(a \cdot b\) ni hisoblaymiz: \[ 23\times 5=115. \] Javob: \(115\) (B varianti). --- ### 16-misol Berilgan: shar radiusi \[ R=4\sqrt{3}, \] ichiga chizilgan silindr hajmining eng katta qiymatini topishimiz kerak. Silindrni shar ichiga chizishda, silindrning radiusi \(r\) va balandligi \(h\) orasidagi munosabat: \[ h=2\sqrt{R^2-r^2}. \] Silindr hajmi: \[ V=\pi r^2h=2\pi r^2\sqrt{R^2-r^2}. \] Eng katta hajmni topish uchun optimal \(r\) ni aniqlash natijasi, isbotlanganidek: \[ r^2=\frac{2}{3}R^2\quad\text{va}\quad h=\frac{2R}{\sqrt{3}}. \] Shunday qilib, maksimal hajm: \[ V_{\max}=\pi \left(\frac{2}{3}R^2\right)\cdot\frac{2R}{\sqrt{3}}=\frac{4\pi R^3}{3\sqrt{3}}. \] Endi \(R=4\sqrt{3}\) ni qo‘yamiz: \[ R^3=(4\sqrt{3})^3=64\,(3\sqrt{3})=192\sqrt{3}. \] Shunda: \[ V_{\max}=\frac{4\pi (192\sqrt{3})}{3\sqrt{3}}=\frac{768\pi\sqrt{3}}{3\sqrt{3}}=256\pi. \] Javob: \(256\pi\) (C varianti).