Modeling with functions?

I have a word problem that I need some help with. I do not fully understand this concept yet and I'm a bit confused on how to create an equation. Here is the word problem:

A quarter mile track is in the shape of a rectangle with semi-circles at both ends. Write a function for the area inside the track in terms of the length of the straight-away(L).

Update:

We are just supposed to write the equation so we're mainly dealing with variables like (L for length and W for width)

4 Answers

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  • ?
    Lv 5
    8 years ago
    Favorite Answer

    Full solution, with explanation, it is a bit long but fun (to a mathie like me)

    -basic knowledge that you should have beforehand:

    Area of a circle (Ao) = pi * Radius^2

    Area of rectangle (A[]) = Length * Width ;Note that for your setup, width = 2*radius

    A[] = L * 2R ;the fewer variables the better, that's why I got rid of width

    -therefore total area (At) of track is rectangle plus 2 halves (aka a whole) circle

    -Note how the perimeter is given as a quarter-mile!

    Perimeter of circle = pi * (2 * radius)

    Perimeter of rectangle = 2 * (length + width); for this purpose though only 2 lengths are part of the total perimeter (Pt)

    --from here it goes faster, with variables only

    Pt = 2 (L) + 2 (pi R)

    = 2 (L + pi*R) ; but since you are given Pt = 0.25 mi

    0.25/2 = L+pi*R ;which lets you find

    -pi*R = L-0.125

    R = {(L-.125) / (-pi)} ;now the only variable is length, just like the question

    At = A[] + Ao ;to find the total area

    At = L * 2R + pi*R^2 ;then sub in radius equation

    At = L * 2{(L-.125) / (-pi)} + pi * {(L-.125) / (-pi)} ^2

    and there you have it, area as a function of the side length in miles

    -Bonus, just because I'm feeling nice I'll simplify it for you into a more usable form :

    At = (L^2 + 0.015625) / pi ;I still encourage you to try the simplification, to try getting the same answer

  • 8 years ago

    Sounds like there isn't enough information. However:

    The total area is the sum of the areas of: (1) the rectangle and (2) the two semi-discs. So, to find the area of the rectangle, you need its length and width. I am guessing, the width is the diameter of the two semi-discs. The two semi-discs together have the area of a circular disc of that diameter.

  • 8 years ago

    solution:

    [please dont copy only. try to understand first coz i m not providing solution for copy only but for the help in ur learning. so plz dont copy without understanding]

    if r mile be the radius of the semicircle

    so area in sq. mile

    = πr² + 2r.(0.25-2πr)

    = πr² - 4πr² + 0.5r

    = 0.5r - 3πr²

    if L be the straight side or length

    so,

    radius of semicircle

    = (0.25-2L)/(2π)

    so,

    area

    = [(0.25L-2L²)/π] + (0.25-2L)²/(4π)

    hope it would be helpful

  • 8 years ago

    Area of Combined Semi-Circles = Ac

    Area of Rectangular Portion = Al

    Length of Staight-Away = L

    Total Area = A

    Ac = π r²

    Al = L(2r)

    A = Ac + Al

    A = π r² + L(2r)

    A = π r² + 2Lr

    A = r(πr + 2L)

    ¯¯¯¯¯¯¯¯¯¯¯¯

     

    Source(s): 8/20/12
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