Problem :
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 |
Problem :
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
θ
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) = | |||
sec^{2} θ(t) = | |||
Rewrite sec as the reciprocal of the cos: | |||
= | |||
Now plug in the particular values: | |||
= (- 20) | |||
= - 2 radians per second |
Problem :
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. | |||
-3 = | |||
= - feet per second |
Problem :
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} | ||
= h(t)^{2} |
4 | = (5)^{2} | ||
= cm per second |