প্রশ্নে বলা হচ্ছে যে, দুটি মোমবাতি একটি সুষম গতিতে 3 ঘণ্টায় ও অপরটি ও 4 ঘণ্টা পুরোপুরি জ্বলে শেষ হয়। কখন দুটি মোম একত্রে জ্বালালে বিকাল 4 টার সময় একটির দৈর্ঘ্য অপরটির দ্বিগুণ থাকবে? 

$$\begin{gathered} Let l be length of each candle and x be the number of hours before 4pm. \\ \therefore Length of 1^{st} candle burnt \left(\frac{1}{3}\right)x \\ \therefore Length of 2^{nd} candle burnt \left(\frac{1}{4}\right)x \\ \therefore Remaining of 1^{st} candle=\left(l-\frac{lx}{3} \right)at 4 pm \\ \& Remaining of 2^{nd} candle= \left(l-\frac{lx}{4}\right)at 4 pm \\ According to question, \\ l - \frac{lx}{4}= 2\left(l-\frac{lx}{3}\right) \\ \Rightarrow l\left( 1 - \frac{x}{4}\right)=2l\left(1-\frac{x}{3}\right) \\ \Rightarrow 1 - \frac{x}{4}=2\left(1-\frac{x}{3}\right) \\ \Rightarrow 1-\frac{x}{4}=2-\frac{2x}{3} \\ \Rightarrow \frac{2x}{3}-\frac{x}{4}=2-1 \\ \Rightarrow \frac{4x-3x}{12}=1 \\ \Rightarrow \frac{5x}{12}=1 \\ \therefore x=\frac{12}{5}=2\frac{2}{5}hours= 2 hours and 24 minutes \\ \therefore Required times= 4-2 hours 24 minutes= 1 hour 36 minutes=1 : 36 pm \end{gathered}$$

(ans)

প্রশ্নে বলা হচ্ছে যে, দুটি মোমবাতি একটি সুষম গতিতে 3 ঘণ্টায় ও অপরটি ও 4 ঘণ্টা পুরোপুরি জ্বলে শেষ হয়। কখন দুটি মোম একত্রে জ্বালালে বিকাল 4 টার সময় একটির দৈর্ঘ্য অপরটির দ্বিগুণ থাকবে? 

$$\begin{gathered} Let l be length of each candle and x be the number of hours before 4pm. \\ \therefore Length of 1^{st} candle burnt \left(\frac{1}{3}\right)x \\ \therefore Length of 2^{nd} candle burnt \left(\frac{1}{4}\right)x \\ \therefore Remaining of 1^{st} candle=\left(l-\frac{lx}{3} \right)at 4 pm \\ \& Remaining of 2^{nd} candle= \left(l-\frac{lx}{4}\right)at 4 pm \\ According to question, \\ l - \frac{lx}{4}= 2\left(l-\frac{lx}{3}\right) \\ \Rightarrow l\left( 1 - \frac{x}{4}\right)=2l\left(1-\frac{x}{3}\right) \\ \Rightarrow 1 - \frac{x}{4}=2\left(1-\frac{x}{3}\right) \\ \Rightarrow 1-\frac{x}{4}=2-\frac{2x}{3} \\ \Rightarrow \frac{2x}{3}-\frac{x}{4}=2-1 \\ \Rightarrow \frac{4x-3x}{12}=1 \\ \Rightarrow \frac{5x}{12}=1 \\ \therefore x=\frac{12}{5}=2\frac{2}{5}hours= 2 hours and 24 minutes \\ \therefore Required times= 4-2 hours 24 minutes= 1 hour 36 minutes=1 : 36 pm \end{gathered}$$

(ans)

We get the following conditions,$x\ge 0,y\ge 0,x+y=5,x\le 2,y\le 4$
As the maximum value of x is 2, 
So the values of x may be x = 0, 1, 2.
And as the maximum value of y is 4,
So the values of y may be y = 0, 1, 2, 3, 4.
And according to another condition x + y = 5.
This also approves the first condition because here x+y=2+3 = 5. 

That mean's if we take x = 2 and y = 3,
Then x+y=5 equation will be true.
From here, we get the maximum value of x = 2 and
The maximum value of y = 3.
The maximum value of z= 6x+2y 

$$\begin{gathered} \Rightarrow z=\left(6\times 2\right)\left(2\times 3\right) \\ \Rightarrow z=12+6 \\ \therefore z=18 \end{gathered}$$