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Please Help Pre-Cal

  1. Oct 23, 2005 #1
    Please Help Pre-Cal !!

    1.A rectangle has one vertex on the line y=L-Fx,x>0 , another at the origin, one on the positive x-axis, and one on the positive y-axis. Find the largest area A that can be enclosed by the rectangle. Show all your work and include a sketch with labels of all important features. (Hint: Start by drawing a diagram, label the base of the rectangle as x, and write the area, A, as a function of x. …)
    Where F = 5, L =4

    SOL : A = f(x)= x * (-5x+4) ; the x is the base while y(x)=-5x+4 is the
    height.
    and the domain of x is 0<x<(4/5)
    To get critical point x= 4/10 which is
    where the maximum occurs at y= .8 ( took the derivative )



    2.A drum in the shape of a right circular cylinder is required to have a volume of cubic centimeters. The top and bottom are made of material that costs F¢ per square centimeter; the sides are made of material that costs L¢ per square centimeters. (Hint: The formula for the volume of a right circular cylinder is V=pie r^2 , where r is the radius of the circular base and h is the height of the cylinder. The surface area of the sides can be determined by cutting the cylindrical shell vertically and flattening it out to get a rectangle whose dimensions can be determined.)


    a)Express the total cost C of the material as a function of the radius r of the cylinder.
    b)What is the cost if the radius is 25 cm?
    c)Graph C=C(r) . Using the graph, for what value of r, approximately, is the cost C least?
    Where F = 5, L =4

    SOl : C= L(2*pi*r*h)+F(2*pi*r^2);
    C(r) = L(2*pi*r*(2*pi*r/sqrt(3))+F(2*pi*r^2); Here height is
    expressed in terms of r . The ratio used to find h in terms of r is
    30-60-90 degree triangle; (2*pi*r)/(h) = sqrt(3)/1

    3.Design a polynomial function with the following characteristics: degree 6; exactly four real zeros, one of multiplicity 3 at x= -F ; y-intercept at F, behaves like y = -Lx^6 for large values of |x| . Give the formula and a complete graph

    Where F = 5, L =4

    ???


    4.Create a rational function that has the following characteristics: crosses the x-axis at F; touches the x-axis at -L ; has one vertical asymptote at x=-L-1 and another at x=F+1 ; and has one horizontal asymptote, y=F . Give the formula and a complete graph

    Where F = 5, L =4


    SOL: Crosses X-Axis @ g
    touches X-Axis @ -4
    Vertical asymtote X= -4-1 = -5
    Veetical asymtote x= 5+1 = 6
    Horizontal asymtote y=5

    5X 1
    --- X ----
    (X+5) (X-6)
     
  2. jcsd
  3. Oct 24, 2005 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Yes, you done that one correctly.
    Were you given other information you didn't tell us? There is nothing in what you gave here that says h must be any particular function of r.
    (And why do you keep saying "F= 5, L= 4"?
    You understand, do you not that, that there are an infinite number of correct answers for this? The problem says "Design a polynomial function". You are just asked to come up with one like this. Since the polynomial is to have a zero of degree 3 at x= -F, it must have factors (x-(-F))3= (x+F)3. It has just one more real root, lets make that x= 1, for simplicity: it has a factor of (x-1). The other two roots must be complex conjugates. Lets take those to be i and -i: another factor is (x2+ 1). The polynomial must be of the form f(x)= (x+F)3(x-1)(x2+ 1). Taking x= 0, f(0)= F3(-1)(1)= -F3, so far. In order that the y intercept be F, we must insure that f(0)= F and we can do that by multiplying the entire polynomial by -F2:
    f(x)= -F2(x+F)3(x-1)(x2+1).
    "crosses the x-axis at F" (does the problem specify that F= 5 or are you assuming that you can choose whatever F you want?) means that there is a factor of (x-F) in the numerator. "touches the x-axis at -L" (I assume "touches" implies that it does not cross the axis) means that there must be a factor of (x-(-L))2= (x+ L)2. To have vertical asymptotes at -L-1 and F+1, there must be factors of (x+L+1) and (x-F-1) in the denominator. So far the numerator has degree one more than the denominator. To make sure the horizontal asymptote is F, we must put another x in the denominator- we can do that without introducing another vertical asymptote putting another factor of either (x+L+1) or (x-F-1) in the denominator and then by multiplying the entire fraction by F:
    [tex]f(x)= F\frac{(x-F)(x+L)^2}{(x-F-1)^2(x+L+1)}[/tex] is one possible answer.
     
  4. Oct 24, 2005 #3
    thanks very much

    Thanks very much for your help.

    No I wasn't given any other information on cylinderical prob..

    and how do I draw a graph for prob 3 & 4. The one with polynomials & Rational questions.

    thanks for your help
     
  5. Oct 24, 2005 #4
    modified question 2

    I am sorry I missed some parts...

    Original Question : A Drum in the shape of a right circular cylinder is required to have a volume of 5000 cubic centimeters. The top and bottom are made of material that costs 5 cents per square centimeter, the sides are made of material that cost 4 cents per sq centimeters. (Hint: The formula for the volume of a right circular cylinder is V=Pie r^2 h, where r is the radius of the circular base and h is the height of the cylinder. The surface are of the sides can be determined by cutting the cylinderical shell vertically and flattening it out to get a rectangle whose dimensions can be determined.)

    a) Express the total cost C of the material as a function of the radius r of the cylinder.
    b) What is the cost if the radius is 25 cm?
    c) Graph C=C(r). Using the graph, for what value of r, approximately, is the cost C least?

    Sol : C = 4 ( 2pie r h ) + 5 (2pie r ^ 2 )
    C(r) = 4 ( 2pie r ( 2 pie r/sroot 3 )) + 5 ( 2pie r^ 2)

    2pie r / h = sq root 3 / 1


    that's all I soved so far...I am lost, please help !!

    thanks for ur help
     
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