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Examples of finding Maxima and Minima

Let us look at two examples of finding maxima and minima of functions.

 

Example 1: A juice manufacturer wants to minimise the amount of plastic needed to make a cylindrical bottle of 500 ml capacity. Given 500 ml = 500 cm3, find the radius of the bottle with minimum surface area.

Volume of a cylinder = pi{r^2}h

Therefore pi{r^2}h = 500

Rearranging the above, we get h = 500/{pi{r^2}}

Surface area of a cylinder S = 2pi{rh}2pi{r^2}

S = 2pi{r}~{(500/{pi{r^2})}}2pi{r^2}

1000/r2pi{r^2}

dS/dr-1000/{r^2}4pi{r}

For minimum value

dS/dr = 0   and    {d^2S}/{dr^2} > 0

When dS/dr = 0, we have

1000/{r^2} = 4pi{r}

r3 =  1000/{4pi}

r =  root{3}{1000/{4pi}}   =  4.30

{d^2S}/{dr^2}  =  2000/{r^3}4pi

= 37.72  > 0  (r = 4.30 from above)

So at r = 4.3 cm, the surface area will be minimum.

And that surface area S = 1000/4.32pi{(4.3)^2}

= 348.73 cm2

 

Example 2: Sand is tipped from a truck onto a pile. The rate, R kg/sec at which the sand is flowing is given by the expression

R = 100t – t3, for 0 ≤ t ≤ T where t is the time in seconds after the sand begins to flow.

What is the largest value of T for which the expression R is physically possible, and find the maximum rate of flow of the sand?

R ≥ 0 since you can’t untip the sand, so let us examine the value of t for which 100t – t3 ≥ 0

t(100 − t²) ≥ 0

t(10 − t)(10 + t) ≥ 0

This is a cubic curve. By substituting the values between -10, 0 and 10, we get the curve

finding maxima and minima

 

The curve is above x-axis in the range 0 ≤ t ≤ 10

Since t cannot be negative, 0 ≤ t ≤ 10 is the solution, and the largest value of t is 10 seconds.

Maximum rate of flow of sand = dR/dt when {dR/dt}~=~0 and {d^2R}/{dt^2}~<~0

{dR/dt} = 100 – 3t2  = 0

3t2  = 100

t = sqrt{100/3}   =  10/sqrt{3}  (t > 0)

{d^2R}/{dt^2}  = -6t  = -6 x 10/sqrt{3}  < 0 hence maximum

The maximum rate of flow is when t = 10/sqrt{3}

R = 100 x 10/sqrt{3}  –  {(10/sqrt{3})^3}

1000/sqrt{3}  –  {1000/{3sqrt{3}}}

= {(3000~-~1000)}/{3sqrt{3}}

= {2000}/{3sqrt{3}}

= 384.90 kg/sec