David and Angela start at the same point. At time
t = 0
, Angela starts running 30ft/sec
north, while David starts running 40ft/sec east. At what rate is the distance between them
increasing when they are 100 feet apart?
|2(80)(40) + 2(60)(30)||=||2(100)|
|=||50 feet per second|
Sophia is sitting on the ground 10 feet from the spot where a hot air balloon is about to
land. She is watching the balloon as it travels at a steady rate of 20 feet per second
towards the ground. If
is the angle between the ground and her line of sight to
the balloon, at what rate is this angle changing at the instant the balloon hits the ground?
|tan θ(t) =|
|sec2 θ(t) =|
|Rewrite sec as the reciprocal of the cos:|
|Now plug in the particular values:|
|= (- 20)|
|= - 2 radians per second|
Indy is 6 feet tall and is walking at a rate of 3 feet per second towards a lamppost that is
18 feet tall. At what rate is his shadow due to the lamppost shortening when he is 6 feet
from the base of the lamppost?
|By similar triangles,|
|= - 3, since this quantity is decreasing.|
|= - feet per second|
Water is being poured into an inverted cone (has the point at the bottom) at the rate of 4
cubic centimeters per second. The cone has a maximum radius of 6cm and a height of 30
cm. At what rate is the height increasing when the height is 3cm?
|V(t) = Π r(t) h(t)|
|r = h|
|V(t)||= Π h(t)3|
|= cm per second|