Parabola(?) Question

• Jan 20th 2009, 01:37 AM
requal
Parabola(?) Question
The product of two positive numbers is 72. If one of the numbers is x,
1. Show that the sum of one number and twice the other can be written as $\displaystyle S=x+\frac{144}{x}$( I can do this part)
2.Give a neat sketch of $\displaystyle S=x+\frac{144}{x}$ for x>0. This is the part I have trouble- i cant tell what sort of a graph it is. The answers show a parabola as the solution but I don't know why it is a parabola
3. What is the size of two numbers which will make the value of S least. (again, cant do this one)
• Jan 20th 2009, 02:03 AM
AlvinCY
2.Give a neat sketch of S=x+\frac{144}{x} for x>0. This is the part I have trouble- i cant tell what sort of a graph it is. The answers show a parabola as the solution but I don't know why it is a parabola

$\displaystyle S=x+\frac{144}{x}$
$\displaystyle Sx=x^2+144$
$\displaystyle x^2-Sx+144=0$

Is it apparent now why this is a parabola? Remember, the SUM is a fixed number, so it can be treated as a constant (Giggle)

3. What is the size of two numbers which will make the value of S least. (again, cant do this one)

To make S the least, it's the same as MINIMISING S, minimisation problems are achieved through calculus, and is obtained by solving $\displaystyle \frac{dS}{dx}=0$

$\displaystyle S=x+\frac{144}{x}$

$\displaystyle \frac{dS}{dx}=0$
$\displaystyle 1-\frac{144}{x^2}=0$
$\displaystyle x^2-144=0$
$\displaystyle x^2=144$
$\displaystyle x=\pm12$

BUT... $\displaystyle x$ is a POSITIVE number (given in the question), so $\displaystyle x=12$, the other number being $\displaystyle \frac{72}{x}=6$.