# 'L varies jointly as a and square root of b, and L = 72 when a = 8 and b = 9. Find L when a = 1/2 and b = 36? Y varies jointly as the cube of x and the square root of w, and Y = 128 when x = 2 and w = 16. Find Y when x = 1/2 and w = 64?

Jul 24, 2017

$L = 9 \text{ and } y = 4$

#### Explanation:

$\text{the initial statement is } L \propto a \sqrt{b}$

$\text{to convert to an equation multiply by k the constant}$
$\text{of variation}$

$\Rightarrow L = k a \sqrt{b}$

$\text{to find k use the given conditions}$

$L = 72 \text{ when "a=8" and } b = 9$

$L = k a \sqrt{b} \Rightarrow k = \frac{L}{a \sqrt{b}} = \frac{72}{8 \times \sqrt{9}} = \frac{72}{24} = 3$

$\text{ equation is } \textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{L = 3 a \sqrt{b}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{when "a=1/2" and "b=36}$

$L = 3 \times \frac{1}{2} \times \sqrt{36} = 3 \times \frac{1}{2} \times 6 = 9$

$\textcolor{b l u e}{\text{-------------------------------------------------------}}$

$\text{Similarly}$

$y = k {x}^{3} \sqrt{w}$

$y = 128 \text{ when "x=2" and } w = 16$

$k = \frac{y}{{x}^{3} \sqrt{w}} = \frac{128}{8 \times 4} = \frac{128}{32} = 4$

$\text{equation is } \textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = 4 {x}^{3} \sqrt{w}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{when "x=1/2" and } w = 64$

$y = 4 \times {\left(\frac{1}{2}\right)}^{3} \times \sqrt{64} = 4 \times \frac{1}{8} \times 8 = 4$