Write the following joint variation using the symbol \(\propto\):
\(x\) varies directly as \(t\) and \(y\) to the power of four and varies inversely as \(r\) and square root of \(s\).

(A)   \(x\propto {\dfrac{t\sqrt{y}}{rs^4}}\)

(B)   \(x\propto {\dfrac{ty^4}{r\sqrt{s}}}\)

(C)   \(x\propto {\dfrac{yt^4}{r\sqrt{s}}}\)

(D)   \(x\propto {\dfrac{t^4y^4}{\sqrt{rs}}}\)

Write an equation for the joint variation based on the information given:
\(z\) varies directly as \(r\) to the power of three and inversely as square root of \(s\) and square root of \(p\) when \(z=5\) and \(r=3,\text{ }s=9,\text{ }p=36\).

(A)   \(z=\dfrac{2r^3}{\sqrt{ps}}\)

(B)   \(z=\dfrac{r^3}{2\sqrt{ps}}\)

(C)   \(z=\dfrac{3r^3}{10\sqrt{ps}}\)

(D)   \(z=\dfrac{10r^3}{3\sqrt{ps}}\)

\(R\) varies inversely as the square root of \(S\) and varies directly as the cube of \(T\). It is given that \(R=\dfrac{kT^a}{S^b}\) and \(k\) is a constant, find the values of \(a\) and \(b\).

(A)  \(a=3,\quad b=\dfrac12\)

(B)  \(a=\dfrac12,\quad b=3\)

(C)  \(a=2,\quad b=\dfrac13\)

(D)  \(a=\dfrac13,\quad b=2\)

It is given that \(M\) varies directly as \(N\) and

inversely as \(P\).

If \(M=20\) when \(N=12\) and \(P=3\),

find the equation relating \(M\), \(N\) and \(P\).

(A)  \(M=4NP\)

(B)  \(M=5NP\)

(C)  \(M=\dfrac{4N}{P}\)

(D)  \(M=\dfrac{5N}{P}\)

Given \(V\) varies directly as \(t\), and inversely as the square of \(r\).

If \(V= kt^p r^q\), where \(k\) is a constant, find the value of \(p\) and of \(q\).

(A)  \(p = 1, \space q = 2\)

(B)  \(p = -1, \space q = 2\)

(C)  \(p = 1, \space q = -2\)

(D)  \( p = -1, \space q = -2\)