# Minimum Distance from Point to Curve

• Mar 25th 2010, 05:06 PM
asemh
Minimum Distance from Point to Curve
Hi
I have been attempting this question, but i do no know how to continue...

Determine the minimal distance from the point (0,4) to the curve y = 8 - x^2

I know the distance formula is $\displaystyle D = \sqrt{(x_2-x_1)^2 + (y_2 - y_1)^2}$

after plugging in the x1 and y1 i got $\displaystyle D = \sqrt{x^2 + (y-4)^2}$

What do i do next? Am i doing a step wrong?

Thanks
• Mar 25th 2010, 05:17 PM
skeeter
Quote:

Originally Posted by asemh
Hi
I have been attempting this question, but i do no know how to continue...

Determine the minimal distance from the point (0,4) to the curve y = 8 - x^2

I know the distance formula is D = sqrt((x2-x1)^2 + (y2 - y1)^2)

after plugging in the x1 and y1 i got = sqrt(x^2 + (y-4)^2)

What do i do next? Am i doing a step wrong?

Thanks

$\displaystyle d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

$\displaystyle (x_2,y_2) = (x,8-x^2)$

$\displaystyle (x_1,y_1) = (0,4)$

$\displaystyle d = \sqrt{(x-0)^2 + [(8-x^2) - 4]^2}$

clean up the expression underneath the radical.

since you posted this in the precalculus and not the calculus section, I assume you are not familiar with using the methods of calculus for optimization.
what method(s) have you been using in class for problems like this?
• Mar 25th 2010, 05:19 PM
asemh
We usually get a formula, derive it, then solve for x
• Mar 25th 2010, 05:42 PM
skeeter
Quote:

Originally Posted by asemh
We usually get a formula, derive it, then solve for x

do you mean take the derivative?
• Mar 25th 2010, 05:46 PM
asemh
Yes

BTW, When i cleaned up the equation i got

$\displaystyle d = \sqrt{ x^4 - 7x^2 + 16}$

is that correct?
• Mar 25th 2010, 06:00 PM
skeeter
Quote:

Originally Posted by asemh
Yes

BTW, When i cleaned up the equation i got

$\displaystyle d = \sqrt{ x^4 - 7x^2 + 16}$

is that correct?

fine ... note that if you minimize $\displaystyle (x^4-7x^2+16)$ , you also minimize its square root.
• Mar 25th 2010, 06:02 PM
asemh
Do you mean when i derive it?

So it will become like 0.5(x^4 - 7x^2 + 16)^-0.5 (4x^3 - 7) ?
• Mar 25th 2010, 06:22 PM
skeeter
Quote:

Originally Posted by asemh
Do you mean when i derive it?

So it will become like 0.5(x^4 - 7x^2 + 16)^-0.5 (4x^3 - 7) ?

finish it.
• Mar 26th 2010, 04:38 AM
HallsofIvy
Quote:

Originally Posted by asemh
Do you mean when i derive it?

So it will become like 0.5(x^4 - 7x^2 + 16)^-0.5 (4x^3 - 7) ?

Skeeter also said "note that if you minimize $\displaystyle x^4- 7x^2+ 16$ , you also minimize its square root.[quote]. Its derivative is $\displaystyle 4x^3- 14x$. Where is that 0?