Can You Spot the Error in This Calculus Problem Solution?

[DeusAbscondus]

*(Incidentally, Is there a collective term for the expression "maxima and minima problems", like, for instance: "optima problems" or "optimizing problems"?)

Main question:

Could someone take a look at my working out, attached as geogebra screenshot, for the following problem:

If the time taken from A to B is given by
$$T=\frac{\sqrt{100^2+x^2}}{3}+100-\frac{x}{5}$$
find a value for x which minimizes time taken for the journey.

Thanks,
deusabs

 

[MarkFL]

At the point where you have:

$\displaystyle 25x^2=9x^2+9(100^2)$

You want to finish with:

$\displaystyle 16x^2=90000$

$\displaystyle x^2=5625$

Taking the positive root, we have:

$\displaystyle x=75$

edit: These types of problems are commonly referred to as optimization problems.

 

[DeusAbscondus]

At the point where you have:

$\displaystyle 25x^2=9x^2+9(100^2)$

You want to finish with:

$\displaystyle 16x^2=90000$

$\displaystyle x^2=5625$

Taking the positive root, we have:

$\displaystyle x=75$

edit: These types of problems are commonly referred to as optimization problems.

Thanks kindly Mark, for putting me out of my misery.
Regs,
DeusAbs

 

[Sudharaka]

*(Incidentally, Is there a collective term for the expression "maxima and minima problems", like, for instance: "optima problems" or "optimizing problems"?)

Main question:

Could someone take a look at my working out, attached as geogebra screenshot, for the following problem:

If the time taken from A to B is given by
$$T=\frac{\sqrt{100^2+x^2}}{3}+100-\frac{x}{5}$$
find a value for x which minimizes time taken for the journey.

Thanks,
deusabs

Hi DeusAbscondus, :)

In your attachment you have written,

\[25x^2=9x^2+9(100^2)\]

\[\Rightarrow 5x=3x+300\]

So you have taken the square root of both sides and assumed that, \(\sqrt{9x^2+9(100^2)}=\sqrt{9x^2}+\sqrt{9(100^2)}\) which is incorrect.

Kind Regards,
Sudharaka.