a151t2au - M ATH 151 Answers to Test 2 Autumn 2009 1 A...

MATH 151
Answers to Test 2
Autumn 2009
1. A baseball diamond is a square with sides 90 feet long. A baseball player ( P in the
figure below) is advancing from the second base ( S ) to the third base ( T ) and the umpire
at the home plate ( H ) is watching him. Let θ be the angle between the third baseline
( TH ) and the line of sight from the umpire to the player ( HP ). How fast is θ changing
when the runner is 30 feet from the third base and running at 24 feet per second?
Let x be the distance PT from the player to the third
base in feet. In the triangle HTP , TH = 90 ft is
constant. The angle PTH = θ and x are changing.
tan θ = x/ 90; so 90 tan θ = x . Di ff erentiate both
sides with respect to time we get 90 sec
2
θ
d θ
dt
=
dx
dt
.
When x = 30, tan θ = 30 / 90 = 1 / 3, and so
sec
2
θ = 1 + tan
2
θ = 1 + (1 / 3)
2
= 10 / 9.
As
dx
dt
= 24, 90 ×
10
9
d θ
dt
= 24.
Hence
d θ
dt
=
24
100
.
Answer : 0 . 24 radians per second
H
F
S
T
P
x
90
θ
2. Find the equation of the tangent line to the graph of the equation at the specified point.