Expert: Sherman D. - 9/19/2005

Question
Thanks for your help in all of this.  The u is a point at the end of the vertical line where it meets the x axis from the diagonal line.  this line is from a parametric equation x=1+t, y=-12t+6.  There is no length of the line sorry.  Does this help at all. -------------------------
Followup To
Question -
Sorry about all the confusion here and I appreciate your time.  If I can prob best explain the drawing it might be easier.  Here goes:

Draw the usual x and y axis.  On the y axis draw a line diagonaly to meet the x axis.  the point on the y axis is labelled b and the one on the x axis is labelled a.  From this diagonal line draw a vertical one to meet the x axis at point u.  The area to the left of this is now shaped as a trapezium and I am to find an expression for this area A.  The equation of the line in non parametric form is y=-12x+18 and I calculated the points a and b to be 1.5 and 18 respectively.

From all of this I am looking for help in:

a.  Determine an expression for the area A in terms of u
b.  Find the completed square form of the above expression
c.  By using the expression in b above I need to find the value of u which gives the max value of A
d.  For what value of u is the value of A equal to half of the max area.  Express answer in surd form.

Thanks for your help.-------------------------
Followup To
Question -
Just to make doubly sure that I do indeed have the right expression I worked it out from the following diagram which forms part of the question.  It is as follows and i hope it comes out ok.

y ¦
 ¦
b ¦
 |   |     |        |      |   |  A   |     |      |       |      |        -----------------------
O       u    a

I have to determine an expression for the trapezium shown as Area A.
-------------------------
Followup To
Question -
I need to complete the square on the following expression and hoped you could help.  The expression is:

(1/2)*u(19.5-u)

The parenthasis around the 1/2 is given to make the expression easier to understand.

Thanks for your help as I have to turn this question in tomorrow.
Answer -
if you have this written right, then

(1/2)u * (19.5 - u)

(u/2)((39/2) - u)

(39u/4) - ((u^2)/2)

(39u - 2u^2)/4 = 0

multiply both sides by 4

39u - 2u^2 = 0

divide everything by 2

19.5u - u^2 = 0

or

-u^2 + 19.5u = 0

divide everything by -1

u^2 - 19.5u = 0

u^2 - ((39u)/2) = 0

find half of (39/2), square it, and add to both sides

half of (39/2) is (39/4), square it and you get (1521/16), and add to both sides

u^2 - ((39u)/2) + ((1521u)/16) = (1521/16)

factor left side into a perfect square

(u - (39/4))^2 = (1521/16)

sqrt both sides

u - (39/4) = ±(39/4)

add (39/4) to both sides

u = (39/4) ± (39/4)

u = 0 or (39/2)

if you meant something else besides (1/2)u(19.5 - u), then let me know.
Answer -
y ¦
 ¦
b ¦
 |   |     |        |      |   |  A   |     |      |       |      |        -----------------------
O       u    a

can you explain it in terms of what each letter stands for.

Is "O" to "a" suppose to be the base of the whole triangle figure. Is "b" to "O" suppose to be the length of the triangle. Is "A" suppose to represent the Area of the shape to be found. Is "u" suppose to stand for the line between "O" and "a" or is it suppose to be the distance between "O" and "a". Are the letters suppose to represent points on the graph or lines. Also does the "y" represent the line from "O" to "y"

is "u" halfway between "O" and "a". If so, i should be able to easily tell you how to solve this problem.

For ex:

-A
-|b|-\c
-|--------
C--a--B

The capital letters are to represent the points.
the lowercase letters are to represent the lines.

If you can, if each letter is a point on the graph, just let me know and i will help.   
Answer -
a. Determine an expression for the area A in terms of u
b. Find the completed square form of the above expression
c. By using the expression in b above I need to find the value of u which gives the max value of A
d. For what value of u is the value of A equal to half of the max area. Express answer in surd form.

I can tell you this, distance from "b" to "a" is (3/2)sqrt(145)

from "O" to "b" is 18
from "O" to "a" is (3/2)

as you already stated.

Total Area of the Whole Triangle is

(27/4)sqrt(109)

Here is my delimma, the line that "u" is connected with. I don't have a length or name of the line. I can find everything else, but that line is my problem.

is Point "u" suppose to be equidistant, or is it just a point on the graph.

if i could find the lenght of the line, i could find the area of the smaller triangle and subtract it from the area of the bigger triangle. and along with the length of the line, i could give you everything in the turns of "u".

Here is about what i can give you

i will use "y" for the unknown point on the diagonal line.

((3/2),0) and (u,y)

D = sqrt((u - (3/2))^2 + y^2)
D = sqrt(((u - (3/2))(u - (3/2)) + y^2)
D = sqrt((u^2 - (3/2)u - (3/2)u + (9/4)) + y^2)
D = sqrt(u^2 - 3u + (9/4) + y^2)

Area = (b/4)sqrt(4a^2 - b^2)
Area = (y/4)sqrt(4(u^2 - 3u + (9/4) + y^2) - y^2)
A = (y/4)sqrt(4u^2 - 12u + 9 + 4y^2 - y^2)
y = (y/4)sqrt(4u^2 - 12u + 9 + 3y^2)
y = (y/8)sqrt(4u^2 - 12u + 9 + 3y^2)

(27/4)sqrt(109) - (y/8)sqrt(4u^2 - 12u + 9 + 3y^2)

(54sqrt(109) - y(sqrt(4u^2 - 12u + 9 + 3y^2)))/8

or

(1/4)h(L1 + L2)
(u/4)(18 + L2)

Since i don't know anything about the Line i can't figure out anything else, unless i am missing something.

By the way, the reason i used (1/4)h(L1 + L2) instead of (1/2)h(L1 + L2), is because we are working with half a Trapezoid.

Answer
Are you saying that the line is (1 + t) to the right and (-12t + 6) up, which the top of the line would give me the point, ((1 + t),(-12t + 6)), is this what you are saying or did you mistype the -12t + 6. If this is what you mean, then i might be able to help. Sorry its taking so long, its just that i need a good idea of the problem along with ALL information that i need to know. For example, in the first question, you didn't include y = -12x + 18, then in the proceeding problem you included that equation, but didn't include the information of the unknown line.

So could you provide me with ALL the info that you are given to find out the answer.