###### The problem

*y is inversely proportional to .*

*When d =10, y =4*

*d is directly proportional to *

*When x = 2, d = 24 *

*Find a formula for y in terms of x . Give your answer in its simplest form. *

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*y is inversely proportional to .*

*When d =10, y =4*

*d is directly proportional to *

*When x = 2, d = 24 *

*Find a formula for y in terms of x . Give your answer in its simplest form. *

$y\alpha \frac{1}{{d}^{2}}\Rightarrow y=\frac{k}{{d}^{2}}$$4=\frac{k}{{10}^{2}}=4x100=k$ ( given y = 4 and d = 10 )

$y=\frac{400}{{d}^{2}}\to (1$)

b) $d\alpha {x}^{2}\Rightarrow d=k{x}^{2}$

$\begin{array}{l}d=24,x=2\\ 24=k{\left(2\right)}^{2}\Rightarrow 24=4k\end{array}$

$\begin{array}{l}K=\frac{24}{4}=6\\ d=6{x}^{2}\to \left(2\right)\end{array}$

$y=\frac{400}{{d}^{2}};d=6{x}^{2}$ ( substitute $d=6{x}^{2}$ in equation 1 )

$y=\frac{400}{6{x}^{2}}\Rightarrow \frac{200}{3{x}^{2}}$

$y=\frac{200}{3{x}^{2}}$