DSTA asked in Science & MathematicsMathematics · 8 years ago

# 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)

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Lv 5
8 years ago

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,

= (0.25-2L)/(2π)

so,

area

= [(0.25L-2L²)/π] + (0.25-2L)²/(4π)

• 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)

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Source(s): 8/20/12