Question

A variable quantity y is equal to sum of two quantities, one of which varies directly as x and the other varies inversely as x. If y= 11 when x= 1 and y = 13 when x =2, find y when x = 3.​

Answer 1

$\fontsize{18}{10}\textbf{\textup{The required value of y when x is 3 is 17.}}$

Step-by-step explanation:  Given that a variable quantity y is equal to sum of two quantities, one of which varies directly as x and the other varies inversely as x.

Also, y = 11 when x = 1 and y = 13 when x =2.

We are to find the value of y when x = 3.

Let p and q be the two quantities such that  y = p + q.

According to the given information, we have

$p\propto x\\\\\Rightarrow p=kx,~~\textup{where k is a proportionality constant}$

and

$q\propto \dfrac{1}{x}\\\\\\\Rightarrow q=\dfrac{h}{x},~~\textup{where h is another proportionality constant}.$

So, we have

$y=kx+\dfrac{h}{x}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

Given that y = 11 when x = 1.

So, equation (i) implies

$11=k\times 1+\dfrac{h}{1}\\\\\Rightarrow k+h=11\\\\\Rightarrow k=11-h~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)$

And y = 13 when x = 2. Equation (i) implies

$13=2k+\dfrac{h}{2}\\\\\Rightarrow 4k+h=26\\\\\Rightarrow 4(11-h)+4=26~~~~~~~~~~~~[\textup{Using equation (ii)}]\\\\\Rightarrow 44-3h=26\\\\\Rightarrow 3h=18\\\\\Rightarrow h=6.$

From equation (ii), we get

$k=11-6=5.$

So, from equation (i), we get

$y=5x+\dfrac{6}{x}.$

Therefore, when x = 3, then the value of y is

$y=5\times3+\dfrac{6}{3}=15+2=17.$

Thus, the required value of y when x is 3 is 17.

Learn more :  A variable quantity y is equal to sum of two quantities, one of which varies directly  x and the other varies inversely as x. If y = 11 when x = 1 and y = 13 when x = 2, find  y when x = 3.​

LInk :https://brainly.in/question/13920090.