CK-12-Pre-Calculus Concepts

10.5. Applications of Parametric Equations http://www.ck12.org

clockwise. Write parametric equations that model Nikki’s position over time. Where will Nikki be three minutes
from now?
Answers:

1. A football field is 100 yards or 300 feet. The parametric equations for a football thrown from (300, 0) back to the
   origin at speedvare:

x=−t·v·cos

(π

4

)

+ 300

y=^12 ·(− 32 )·t^2 +t·v·sin

(π

4

)

Substituting the point (0, 0) in for(x,y)produces a system of two equations with two variablesv,t.

0 =−t·v·cos

(π

4

)

+ 300

0 =^12 ·(− 32 )·t^2 +t·v·sin

(π

4

)

You can solve this system many different ways.

t=^5

√ 3

2 ≈^4.^3 seconds,v=^40

√

6 ≈ 97. 98 f t/s
In order for someone to throw a football at a 45◦angle all the way across a football field, they would need to throw
at about 98f t/swhich is about 66.8 mph.

(^981) secf eet· (^36001) hoursec· 52801 milef eet≈ 661. (^8) hourmiles

2. Calculate the velocity of each person and of the wind in feet per second:

67 miles

1 hour ·

1 hour

3600 sec·

5280 f eet

1 mile ≈^98.^27 f t/sec

60 miles

1 hour ·

1 hour

3600 sec·

5280 f eet

1 mile =^88 f t/sec

4 miles

1 hour·

1 hour

3600 sec·

5280 f eet

1 mile ≈^5.^87 f t/sec

The location of Danny’s ball can be described with the following parametric equations (in radians). Note that the
wind simply adds a linear term.

x 1 =−t· 98. 27 ·cos

(π

4

)

+ 300 + 5. 87 t

y 1 =^12 ·(− 32 )·t^2 +t· 98. 27 ·sin

(π

4

)

The location of Johnny’s ball can be described with the following parametric equations.

x 2 =t· 88 ·cos

( 5 π

18

)

+ 5. 87 t

y 2 =^12 ·(− 32 )·t^2 +t· 88 ·sin

( 5 π

18

)