Expert: Scotto - 4/11/2009

Question
QUESTION: 1.) Find the equation of the line that passes thru (5, -8) and will form the largest triangle with the positive x-axis and with the negative y - axis

2.) Find the area bounded by the parabola y = x^2 + 2, y = x - 4 and 6x + 5y + 30 = 0

3.) find the limit of 4x^4*[sin (180/x)]^4 as x increases without bound

4.)Find the volume generated when the area bounded by y = x^2 -2, y = 2 and x = 5 is revolved about x = 6 and about y = 25

5.)

Find the dimensions of the rectangle with the largest area inscribed inside the area bounded by y = 16 - x^2 and y = 0 if one side of the rectangle must coincide with x - axis



ANSWER: 1.) Find the equation of the line that passes thru (5, -8) and will form the largest triangle with the positive x-axis and with the negative y - axis

It must be of the form y+8 = m(x-5) from the point given,
so y = m(x-5) - 8.  When x is 0, y is -5m - 8, which I'll call
y0 = -5m - 8. When y is 0, we have 8 = m(x-5) => 8/m = x-5
=> 5 + 8/m = x.  I'll call this value x0 = 5 + 8/m.

The area is known to be (x0)(y0)/2.  Putting in the values gives
(5 + 8/m)(-5m - 8) = -25m - 80 - 64/m = A for area.

dA/dm = -25 + 64/m^2; setting this to 0 gives m² = 64/25 => m = 8/5.

This says the equation is y+8 = 8(x-5)/5 => 5y + 40 = 8x - 40 =>
5y = 8(x-10) => y = 8(x-10)/5 or y = 8x/5 - 16.



2.) Find the area bounded by the parabola y = x^2 + 2, y = x - 4 and 6x + 5y + 30 = 0

I will send along what they look like.  According to that,
there must be something wrong since the parabola y = x²+2 is above the two lines y = x-4 and y = -(6/5)x - 6.


3.) find the limit of 4x^4*[sin (180/x)]^4 as x increases without bound

This is that same as 4(sin(180y)/y)^4 as y->0.
I don't know whether y is in degrees or radians with the 180 there.
That makes me think that y is in degrees and this converts to
4(sin(pi*y)/y)^4, which is the same as 4pi^4sin((pi*y)/(pi*y))^4,
and the limit of sin(pi*y)/(pi*y) = 1 as y -> 0.


4.)Find the volume generated when the area bounded by y = x^2 -2, y = 2 and x = 5 is revolved about x = 6 and about y = 25

If we are rotating a parabola y = x^2-2 around y=25,
the distance to the curve is 25 - y.
The start of the integral is 2^2-2 = 4-2 = 2.
So we'll do the integral from 2 to 5 of π((25-2)² - (25-x^2-2)²).
Note the
That's - (25-x^2-2)² is (23-x)².
⌠5
⌡2 π(529 –529+46x-x²)dx =
⌠5
⌡2 π(46x-x²)dx = π(23x²-x^3/3) from 2 to 5, which is
π(23(25-4)-(125-8)/3) = π(483 – 39) = 444π.


5.) Find the dimensions of the rectangle with the largest area inscribed inside the area bounded by y = 16 - x^2 and y = 0 if one side of the rectangle must coincide with x - axis

The area of the rectangle is xy.  We know that y is 16 – x².
This would make the area 16x – x^3.
The derivative is 16 – 3x².
To maximize the problem, we set the derivative to 0.
This gives us 16=3x², or x = 4/√3.
That’s 4√3/3, to put the square-root in the top.
This means that y is 16 – 16*3/9 = 16-16/3 = 32/3.
The area is xy, and we know x=4√3/3 and y=32/3,
so the area is 128√3/9, which is roughly 24.63361.



---------- FOLLOW-UP ----------

QUESTION: ah im dont get your solution in number ...  i notice that you let y = 1/x and subtitute it to become 4[(sin180y)/y]^4 as y->0??? pls help me

Answer
y is in radians
-------------------------------
On the question, note that (sin180y)/y is the same as
180*(sin180y)/(180y).  If we are taking the limit as y->0 of
[(sin180y)/y]^4, we can say that's the same as taking the limit as
y-> 0 of 180^4[(sin180y)/(180y)]^4.

Let z=180y.  That's 180^4 { which is 1,049,760,000 } *
lim z->0 sinz/z {see * at bottom}.  It is know that the value of the limit by itself is 1, so the result is 1,049,760,000.

y is in degrees
------------------------
Unless, of course, y is in degrees, in which case the first thing to do is to convert it to radians.  To do this, we multiply by π/180.
This gives sin(πy)/y, which is converted to π*sin(πy)/(πy).  The limit of that {*} can be seen to be π*1 = π.




* lim z->0 sinz/z = 1:
On the sinz, note that the first term in the sinz by Taylor's theorem is z.  Divide this by z and you get 1.  The next term involves a z^3, divided by z gives a term with a z² in it.  The rest of the terms involve higher powers of z.  Let z go to 0, and the first term is all that we have left, which is 1.