4x4 = 16
nswer any four questions:
a) 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.
Answers
\fontsize{18}{10}\textbf{\textup{The required value of y when x is 3 is 17.}}\fontsize1810The 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
\begin{gathered}p\propto x\\\\\Rightarrow p=kx,~~\textup{where k is a proportionality constant}\end{gathered}
p∝x
⇒p=kx, where k is a proportionality constant
and
\begin{gathered}q\propto \dfrac{1}{x}\\\\\\\Rightarrow q=\dfrac{h}{x},~~\textup{where h is another proportionality constant}.\end{gathered}
q∝
x
1
⇒q=
x
h
, where h is another proportionality constant.
So, we have
y=kx+\dfrac{h}{x}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)y=kx+
x
h
(i)
Given that y = 11 when x = 1.
So, equation (i) implies
\begin{gathered}11=k\times 1+\dfrac{h}{1}\\\\\Rightarrow k+h=11\\\\\Rightarrow k=11-h~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)\end{gathered}
11=k×1+
1
h
⇒k+h=11
⇒k=11−h (ii)
And y = 13 when x = 2. Equation (i) implies
\begin{gathered}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.\end{gathered}
13=2k+
2
h
⇒4k+h=26
⇒4(11−h)+4=26 [Using equation (ii)]
⇒44−3h=26
⇒3h=18
⇒h=6.
From equation (ii), we get
k=11-6=5.k=11−6=5.
So, from equation (i), we get
y=5x+\dfrac{6}{x}.y=5x+
x
6
.
Therefore, when x = 3, then the value of y is
y=5\times3+\dfrac{6}{3}=15+2=17.y=5×3+
3
6
=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.
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