Cobby's Methods Questions

[cobby]

And another one please


If , show

I keep getting muddled up with these questions


Thanks

[Flaming_Arrow]

get a common denominator





and sub in y

[cobby]

Hey guys got another question

Taken from 2007 MMC Exam 1


P is the point on the line such that the length of , the line segment from the origin to , is a minimum. Find the coordinates of AND this minimum length.


Thanks guys

[kamil9876]

there are two appraoches, the geometrical and the analytical. Since you're into differentiation lately i'll do the latter:

find formula for distance:

where P=(x,y)
y=10-2x:



-
When OP is at a minimum, is at a minimum since the square root function has no turning points.

differentiating and setting derivative to zero:



I think you can take it from here.

--------------------------------------------------------------------

Geometrical:

Draw the circle with radius OP and centre 0. The line is a tangent of the circle, hence the radial line is perpendicular to the tangent:

Radial line:

y=mx+c

(negative reciprical of gradient of given line)

c=0 since it goes through centre which is origin.

radial line:

Hence the point P is the intersection of 2x+y-10=0 and the radial line, which is easy to find.

[cobby]

Thanks kamil

But where did you get from in the first solution?

[d0minicz]

is that sposed to be -5x^2 - 40x + 100 ?

[kamil9876]

Thanks kamil

But where did you get from in the first solution?

LOl sorry its supposde to be 5x^2-40x+100. (thing inside square root)
I basically had the -3 initially but then forgot to change it in every part where i wrote it.

[cobby]

Another one please


The tangent to the curve of at the point where is parallel to the line . There is a stationary point at . Find the values of


Thanks guys

[kamil9876]

you have to set up some simultaneous equations. The pieces of information provide you with this.
"There is a stationary point at 1,-3" has two pieces of information, the point is on that curve and the derivative is zero at x=1:

first piece of info: -3=a+b+c (1)

second piece of info:

dy/dy=2ax+b
0=2a+b (2)

Now one final piece of info:

at x=2, dy/dy=4:

4=2a(2)+b
4=4a+b (3)

use (2) and (3) to find a and b, then plug into (1) to find c

[GerrySly]

Me thinks

EDIT: Wow, I keep getting beaten to posting help

[Flaming_Arrow]

Answered on MSN

[cobby]

Thanks guys!!

[cobby]

A rectangular block is such that the sides of its base are of length x cm and 3x cm. The sum of the lengths of all its edges is 20cm

show that the volume Vcm^3 is given by

Yes i know, the height is not given hence why i cant get the volume...is there another way of doing this q or is it a misprint??


Thanks guys

[kamil9876]

you can find the height with the additional information of "the sum of edge lengths is 20". WHen apparently missing a piece of information, look at how another piece of information may compensate.

You can see that there are four edges of length x, four of length 3x, and four of length h.

hence:

[dcc]

Hey guys got another question

Taken from 2007 MMC Exam 1


P is the point on the line such that the length of , the line segment from the origin to , is a minimum. Find the coordinates of AND this minimum length.


Thanks guys

For those who are interested, kamil9876's Geometrical approach can be generalised using vector magic.

If we have a line with real coefficients, we can write this in the form .  Now we note that the DIRECTION of the line is given by the vector .  Therefore to find our minimum, we wish to find some such that (Think about it, if is not perpendicular to , then there is always another line you can draw which will be shorter).

We find that

Therefore, we notice that the minimum distance from the origin to a point on the line always occurs when .

(Ignore this if you don't care or aren't interested)