2.8-bo'lim : Kvadrat tenglamalarni qo'llash
3. Ikki kishi 14 soat ichida uyni bo'yashlari mumkin. Odamlardan birining yakka tartibda ishlashi ikkinchisiga uyni bo'yash uchun sarflaganidan 2 soat ko'proq vaqt oladi. Uyni bo'yash uchun har bir kishi alohida ishlasa qancha vaqt ketadi?
Yechim
Birinchidan, A shaxsi ikkita rassomdan tezroq bo'lsin va ruxsat bering t uyni o'zi bo'yash uchun qancha vaqt kerak bo'lsa. Keyin B shaxsi ikki rassomdan sekinroq bo'lsin va shuning uchun bu odamga uyni o'zi bo'yash uchun \(t + 2\) soat kerak bo'ladi.
2-qadam
Birgalikda ishlasa, ular 14 soat ichida uyni bo'yashlari mumkin, shuning uchun biz uyni bo'yash uchun birgalikda ishlash uchun quyidagi so'z tenglamasiga egamiz.
\[\left( \begin{array}{c}{\mbox{Portion of job }}\\ {\mbox{ done by Person A}}\end{array} \right) + \left( \begin{array}{c}{\mbox{Portion of job}}\\ {\mbox{done by Person B}}\end{array} \right) = 1{\mbox{ Job}}\]Biz bilamizki, Ish qismi = Ish tezligi X ish vaqti, shuning uchun bu quyidagi so'z tenglamasini beradi.
\[\begin{align*}\left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{of Person A}}\end{array} \right)\left( \begin{array}{c}{\mbox{Work Time}}\\ {\mbox{of Person A}}\end{array} \right) + \left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{of Person B}}\end{array} \right)\left( \begin{array}{c}{\mbox{Work Time}}\\ {\mbox{of Person B}}\end{array} \right) & = 1\\\left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{of Person A}}\end{array} \right)\left( {14} \right) + \left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{of Person B}}\end{array} \right)\left( {14} \right) & = 1\end{align*}\]3-qadam
Endi bizga har bir odamning ish tezligi kerak bo'lib, biz ularni individual bo'yash vaqtlaridan quyidagicha olishimiz mumkin:
\[\begin{align*}\left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{of Person A}}\end{array} \right)\left( \begin{array}{c}{\mbox{Work Time}}\\ {\mbox{of Person A}}\end{array} \right) = \left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{of Person A}}\end{array} \right)\left( t \right) & = 1\\ & \Rightarrow \hspace{0.25in}{\mbox{Work Rate of Person A = }}\frac{1}{t}\end{align*}\] \[\begin{align*}\left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{of Person B}}\end{array} \right)\left( \begin{array}{c}{\mbox{Work Time}}\\ {\mbox{of Person B}}\end{array} \right) = \left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{of Person B}}\end{array} \right)\left( {t + 2} \right) & = 1 \\ & \Rightarrow \hspace{0.25in}{\mbox{Work Rate of Person B = }}\frac{1}{{t + 2}}\end{align*}\]4-qadam
Bularni 2-bosqichdagi so'z tenglamasiga qo'shib, biz quyidagi tenglamaga erishamiz.
\[\begin{align*}\left( {\frac{1}{t}} \right)\left( {14} \right) + \left( {\frac{1}{{t + 2}}} \right)\left( {14} \right) & = 1\\ \frac{{14}}{t} + \frac{{14}}{{t + 2}} & = 1\end{align*}\]5-qadam
Buni hal qilish uchun biz maxrajlarni tozalash uchun LCD, \(t\left( {t + 2} \right)\) ga ko'paytirishimiz kerakligini bilamiz. Shunday qilib,
\[\begin{align*}14\left( {t + 2} \right) + 14t & = t\left( {t + 2} \right)\\ 28t + 28 & = {t^2} + 2t\\ {t^2} - 26t - 28 & = 0\end{align*}\]Biroz soddalashtirilgandan so'ng, biz hal qilish uchun juda oddiy kvadrat tenglamaga erishamiz. Kvadrat formuladan foydalanib,
\[t = \frac{{ - \left( { - 26} \right) \pm \sqrt {{{\left( { - 26} \right)}^2} - 4\left( 1 \right)\left( { - 28} \right)} }}{{2\left( 1 \right)}} = \frac{{26 \pm \sqrt {788} }}{2}\]6-qadam
Oldingi bosqichlarda olingan ikkita qiymatni o'nli kasrlarga qisqartirish orqali biz 2-bosqichdagi kvadrat tenglamaning quyidagi ikkita yechimiga erishamiz.
\[t = \frac{{26 - \sqrt {788} }}{2} = - 1.0357\hspace{0.25in}\hspace{0.25in}t = \frac{{26 + \sqrt {788} }}{2} = 27.0357\]Tenglamaning birinchi yechimi hech qanday ma'noga ega emas, chunki u salbiy (biz vaqt bilan ishlayapmiz va shuning uchun biz \(t = 0\) dan boshlaymiz deb taxmin qilish xavfsiz!), demak, ikkinchisi bizga kerak bo'lgan javobdir.
Bu shuni anglatadiki, A shaxsi uyni 27,0357 soatda, B shaxsi esa 29,0357 soatda (A shaxsidan ikki soat ko'proq) uyni bo'yashi mumkin.