The value of q varies directly withp2,andq=16whenp=2.What is the value of q whenp=12?

1 answer:

7 0

$\bf \qquad \qquad \textit{direct proportional variation}\\\\
\textit{\underline{y} varies directly with \underline{x}}\qquad \qquad y=kx\impliedby 
\begin{array}{llll}
k=constant\ of\\
\qquad variation
\end{array}\\\\
-------------------------------$

$\bf \textit{q varies directly with p}^2\qquad q=kp^2
\\\\\\
\textit{we also know that }
\begin{cases}
q=16\\
p=2
\end{cases}\implies 16=k2^2\implies \cfrac{16}{2^2}=k
\\\\\\
4=k\qquad therefore\qquad \boxed{q=4p^2}
\\\\\\
\textit{when p = 12, what is \underline{q}?}\qquad q=4(12^2)$

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Step-by-step explanation:

Lagrange Multipliers

It's a method to optimize (maximize or minimize) functions of more than one variable subject to equality restrictions.

Given a function of three variables f(x,y,z) and a restriction in the form of an equality g(x,y,z)=0, then we are interested in finding the values of x,y,z where both gradients are parallel, i.e.

$\bigtriangledown f=\lambda \bigtriangledown g$

for some scalar $\lambda$ called the Lagrange multiplier.

For more than one restriction, say g(x,y,z)=0 and h(x,y,z)=0, the Lagrange condition is

$\bigtriangledown f=\lambda \bigtriangledown g+\mu \bigtriangledown h$

The gradient of f is

$\bigtriangledown f=$

Considering each variable as independent we have three equations right from the Lagrange condition, plus one for each restriction, to form a 5x5 system of equations in $x,y,z,\lambda,\mu$ .