Question

If I vary inversely as ${{\text{d}}^{2}}$and I = 20, when d = 3, then what is the value of I when d = 10?
A. $6$
B. $66\dfrac{2}{3}$
C. $18$
D. $1.8$
E. $12$

Answer 1

Hint: Generate the proportionate relationship between I and d given in the question. Assume a constant of proportionality between the two variables and find its value using the condition given. Thus, using this value of the constant, find the value of I when the value of d is equal to 10.

Complete step by step answer:
It is given that, I is inversely proportional to ${{\text{d}}^{2}}$. Thus, it can be expressed in the form,

$\begin{align}

  & \text{I }\propto \text{ }\dfrac{1}{{{\text{d}}^{2}}} \\

 & \therefore \text{ I = }\dfrac{\text{k}}{{{\text{d}}^{2}}}\text{ (where, k is defined as the constant of proportionality) }...\text{(i)} \\

\end{align}$

Now, it is given that, I = 20, when d = 3. Putting these value of I and d in the equation (i), we obtain k as,

$\begin{align}

  & 20\text{ = }\dfrac{\text{k}}{{{3}^{2}}} \\

 & \therefore \text{ k = 20 x 9 = 180} \\

\end{align}$

We need to find out the value of I, when the corresponding value of d is equal to 10. Thus, putting the value of d = 10 and the obtained value of k = 180, we get,

$\begin{align}

  & \text{I = }\dfrac{180}{{{10}^{2}}} \\

 & \therefore \text{ I = }\dfrac{180}{100}\text{ = 1}\text{.8} \\

\end{align}$

Thus, the value of I is 1.8 when the corresponding value of d is equal to 10.

Hence, the correct answer is option D.

Note: One can also find the value of I when d = 10, without explicitly finding out the value of the proportionality constant k. Observe from the equation (i), we can write that
$\begin{align}

  & {{\text{I}}_{1}}\text{ = }\dfrac{\text{k}}{{{\text{d}}_{1}}^{2}}\text{ }....\text{(ii)} \\

 & {{\text{I}}_{2}}\text{ = }\dfrac{\text{k}}{{{\text{d}}_{2}}^{2}}\text{ }....\text{(iii)} \\

\end{align}$

where, $\left( {{\text{I}}_{1}},{{\text{d}}_{1}} \right)$and $\left( {{\text{I}}_{2}},{{\text{d}}_{2}} \right)$are the two corresponding values for the variables I and d. The value of k remains unchanged in both the equations (ii) and (iii) as it is a constant value.

Now, dividing equation (ii) by (iii), we get,

$\dfrac{{{\text{I}}_{1}}}{{{\text{I}}_{2}}}\text{ = }\dfrac{{{\text{d}}_{2}}^{2}}{{{\text{d}}_{1}}^{2}}$

In this equation, putting the values ${{\text{I}}_{1}}\text{ = 20, }{{\text{d}}_{1}}\text{ = 3 and }{{\text{d}}_{2}}\text{ = 10}$, we get,

$\begin{align}

  & \dfrac{20}{{{\text{I}}_{2}}}\text{ = }\dfrac{{{10}^{2}}}{{{3}^{2}}} \\

 & \therefore \text{ }{{\text{I}}_{2}}\text{ = }\dfrac{20\text{ x 9}}{100}\text{ = 1}\text{.8} \\

\end{align}$

Hence, we obtain the same value of ${{\text{I}}_{2}}$without explicitly determining the value of k.