\( h \propto \sqrt{p} \)
\( h = 5.4 \) when \( p = 1.44 \).
Find \( h \) when \( p = 2.89 \).
\( h \propto \sqrt{p} \)
\( h = k\sqrt{p} \)
\( 5.4 = k\sqrt{1.44} \)
\( 5.4 = k(1.2) \)
\( k = 4.5 \)
\( h = 4.5\sqrt{p} \)
When \( p = 2.89 \),
\( h = 4.5\sqrt{2.89} \)
\( h = 4.5(1.7) \)
\( h = 7.65 \)
\( R \propto \frac{1}{d^2} \)
\( R = 10 \) when \( d = 2 \).
Find \( R \) when \( d = 4 \).
\( R \propto \frac{1}{d^2} \)
\( R = \frac{k}{d^2} \)
\( 10 = \frac{k}{2^2} \)
\( 10 = \frac{k}{4} \)
\( k = 40 \)
\( R = \frac{40}{d^2} \)
When \( d = 4 \),
\( R = \frac{40}{4^2} \)
\( R = \frac{40}{16} \)
\( R = 2.5 \)
\( w \propto \frac{1}{\sqrt{x}} \)
\( w = 4 \) when \( x = 4 \).
Find \( w \) when \( x = 25 \).
\( w \propto \frac{1}{\sqrt{x}} \)
\( w = \frac{k}{\sqrt{x}} \)
\( 4 = \frac{k}{\sqrt{4}} \)
\( 4 = \frac{k}{2} \)
\( k = 8 \)
\( w = \frac{8}{\sqrt{x}} \)
When \( x = 25 \),
\( w = \frac{8}{\sqrt{25}} \)
\( w = \frac{8}{5} \)
\( w = 1.6 \)
\( p \propto \frac{1}{(q+4)^2} \)
\( p = 2 \) when \( q = 2 \).
Find \( p \) when \( q = -2 \).
\( p \propto \frac{1}{(q+4)^2} \)
\( p = \frac{k}{(q+4)^2} \)
\( 2 = \frac{k}{(2+4)^2} \)
\( 2 = \frac{k}{36} \)
\( k = 72 \)
\( p = \frac{72}{(q+4)^2} \)
When \( q = -2 \),
\( p = \frac{72}{(-2+4)^2} \)
\( p = \frac{72}{2^2} \)
\( p = \frac{72}{4} \)
\( p = 18 \)
\( y \propto x+5 \)
\( y=4 \) when \( x=-1 \).
Find \( y \) when \( x=11 \).
\( y \propto x+5 \)
\( y = k(x+5) \)
\( 4 = k(-1+5) \)
\( 4 = 4k \)
\( k = 1 \)
Therefore,
\( y = x+5 \)
When \( x=11 \),
\( y = 11+5 \)
\( y = 16 \)
\( V \propto (r+1)^3 \)
\( V = 24 \) when \( r = 1 \).
Find \( V \) when \( r = 2 \).
\( V \propto (r+1)^3 \)
\( V = k(r+1)^3 \)
\( 24 = k(1+1)^3 \)
\( 24 = 8k \)
\( k = 3 \)
\( V = 3(r+1)^3 \)
When \( r = 2 \),
\( V = 3(2+1)^3 \)
\( V = 3(27) \)
\( V = 81 \)
\( y \propto \sqrt{x+5} \)
\( y = 4 \) when \( x = -1 \).
Find \( y \) when \( x = 11 \).
\( y \propto \sqrt{x+5} \)
\( y = k\sqrt{x+5} \)
\( 4 = k\sqrt{-1+5} \)
\( 4 = k\sqrt{4} \)
\( 4 = 2k \)
\( k = 2 \)
\( y = 2\sqrt{x+5} \)
When \( x = 11 \),
\( y = 2\sqrt{11+5} \)
\( y = 2\sqrt{16} \)
\( y = 2(4) \)
\( y = 8 \)
\( m \propto x^3 \)
\( m = 200 \) when \( x = 2 \).
Find \( m \) when \( x = 0.4 \).
\( m \propto x^3 \)
\( m = kx^3 \)
\( 200 = k(2)^3 \)
\( 200 = 8k \)
\( k = 25 \)
\( m = 25x^3 \)
When \( x = 0.4 \),
\( m = 25(0.4)^3 \)
\( m = 25(0.064) \)
\( m = 1.6 \)
\( y \propto \frac{1}{\sqrt{x+1}} \)
\( y = 2 \) when \( x = 8 \).
Find \( y \) when \( x = 99 \).
\( y \propto \frac{1}{\sqrt{x+1}} \)
\( y = \frac{k}{\sqrt{x+1}} \)
\( 2 = \frac{k}{\sqrt{8+1}} \)
\( 2 = \frac{k}{3} \)
\( k = 6 \)
\( y = \frac{6}{\sqrt{x+1}} \)
When \( x = 99 \),
\( y = \frac{6}{\sqrt{99+1}} \)
\( y = \frac{6}{\sqrt{100}} \)
\( y = \frac{6}{10} \)
\( y = 0.6 \)
\( x \propto \sqrt[3]{y} \)
\( x = 6 \) when \( y = 8 \).
Find \( x \) when \( y = 64 \).
\( x \propto \sqrt[3]{y} \)
\( x = k\sqrt[3]{y} \)
\( 6 = k\sqrt[3]{8} \)
\( 6 = 2k \)
\( k = 3 \)
\( x = 3\sqrt[3]{y} \)
When \( y = 64 \),
\( x = 3\sqrt[3]{64} \)
\( x = 3(4) \)
\( x = 12 \)
\( y \propto \sqrt[3]{x+3} \)
\( y = 1 \) when \( x = 5 \).
Find \( y \) when \( x = 340 \).
\( y \propto \sqrt[3]{x+3} \)
\( y = k\sqrt[3]{x+3} \)
\( 1 = k\sqrt[3]{5+3} \)
\( 1 = k\sqrt[3]{8} \)
\( 1 = 2k \)
\( k = \frac{1}{2} \)
\( y = \frac{1}{2}\sqrt[3]{x+3} \)
When \( x = 340 \),
\( y = \frac{1}{2}\sqrt[3]{340+3} \)
\( y = \frac{1}{2}\sqrt[3]{343} \)
\( y = \frac{1}{2}(7) \)
\( y = 3.5 \)