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# The Calculus of a Marathon

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## Amy Rechkemmer

on 27 May 2014

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#### Transcript of The Calculus of a Marathon

The Calculus of Running a Marathon
design by Dóri Sirály for Prezi
Used by many runners

We know it shows location but...

how does it work?

demo

Let's Find Out!
Runner A Calculations
Global Positioning System
Runner A: a = 3 cos(x) + 1
Runner B: a = x/20

Pre-race info:
Both runners start out with a velocity of 0
Runner B oversleeps and comes late to the race, starting the race 200 meters behind the start line
And They're Off!
V = 3 sin(x) + x

X = 3 sin(x) + x dx
X = -3 cos(x) + x^2/2 + c
0 = -3 cos(0) + 0 + c
0 = -3 + c
C = 3

X = -3 cos(x) + x^2/2 +3

A = 3 cos(x) + 1

V = 3 cos(x) + 1 dx
V = 3 sin(x) + x + c

0 = 3 sin(0) + 0 + c
c = 0

V = 3 sin(x) + x

Runner B Calculations
V = 1/40 x^2

X = 1/40 x^2 dx
X = 1/120 x^3 + c
-200 = 1/120 (0)^3 + c
c = -200

X = 1/120 x^3 - 200

A = x/20

V = x/20 dx
V = 1/40 x^2 + c
0 = 1/40 (0)^2 + c
c = 0

V = 1/40 x^2

When Will Runner A Finish?
42, 195 =
-3 cos(x) + x^2/2 +3

0 = -3 cos(x) + x^2/2 – 42, 192

t = 290.49 minutes to finish the marathon

That is close to 5 hours

Marathon = 26.2 miles
26.2 mi = 42, 195 m

To find time it takes to finish the race, we can plug 42, 195 in for X and solve for (x), or time in minutes

When Will Runner B Finish?
42, 195 = 1/120 x^3 – 200

42, 395 = 1/120 x^3

t = 171.99 minutes to finish the marathon

That is close to 3 hours

Marathon = 26.2 miles
26.2 mi = 42, 195 m

To find time it takes to finish the race, we can plug 42, 195 in for X and solve for (x), or time in minutes

Full transcript