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Related Rates: Sphere
Related Topics
Wize University Calculus 1 Textbook > Applications of Differentiation
Related Rates
6 Activities
The volume of a spherical snow ball is decreasing at the rate of
50
50
50
ft
3
/min. What is the rate of change of the area when
r
=
3
r=3
r
=
3
ft? Is the area increasing or decreasing at this instant?
What is the rate of change of the area when
r
=
3
r=3
r
=
3
ft? (enter your answer in ft^2/min, do not include units)
Is the area increasing or decreasing at this instant? (increasing/decreasing)
I don't know
Check Submission
More Related Rates Questions:
Related Rates
An inverted conical tank of water is used as a water trough to hydrate some animals on a farm. However, there is a leak at the bottom of the tip causing water to drip out at a rate of
1000
c
m
3
/
s
1000 \; cm^3 /s
1000
c
m
3
/
s
. The height of the cone is
5
m
5 \; m
5
m
and the diameter at the top of the cone is
4
m
4 \; m
4
m
. How fast is the height of the water decreasing when the water is 200cm high inside the tank. (The volume of a cone is
1
3
π
r
2
h
\frac{1}{3} \pi r^2 h
3
1
π
r
2
h
where
r
r
r
is the radius of the base and
h
h
h
is the height of the cone).
Practice: Related Rates
A plane flies 3 km over a student, moving parallel to the ground at 2km/s. How fast is its distance from the student changing when it has travelled 4 km?
Related Rates
A man running straight at a rate of
10
10
10
ft/sec along a level street passes under a vertically rising balloon when it is
50
50
50
feet high. Assuming that the balloon is rising at a constant rate of
1.5
1.5
1.5
ft/sec, how fast is the distance between the balloon and the running increasing
2
2
2
seconds later?
Related Rates
Suppose there is an isosceles triangle with a constant base of 10cm. If the other two equal sides are increasing at a rate of 2cm/s, at what rate is the area of the triangle increasing when each side is 10cm long?
Practice: Related Rates 2
A cylinder is being flattened so that its volume does not change. If the height of the cylinder is decreasing at 0.4 cm/s, find the rate of change of the radius when the radius is 3 cm and the height is 4 cm.
Related Rates
A bunny and a turtle start from the same place at time
t
=
0
t=0
t
=
0
seconds. The bunny heads east at
5
5
5
m/s. 10 seconds later, the turtle heads south at
25
7
\frac{25}{7}
7
25
m/s and the bunny continues on with it's path. How fast is the distance between this bunny and turtle changing at
t
=
24
t=24
t
=
24
seconds?
Related Rates
An inverted conical tank of water is used as a water trough to hydrate some animals on a farm. However, there is a leak at the bottom of the tip causing water to drip out at a rate of
1000
c
m
3
/
s
1000 \; cm^3 /s
1000
c
m
3
/
s
. The height of the trough is
5
m
5 \; m
5
m
and the diameter at the top of the trough is
4
m
4 \; m
4
m
. (The volume of a cone is
1
3
π
r
2
h
\frac{1}{3} \pi r^2 h
3
1
π
r
2
h
where
r
r
r
is the radius of the base and
h
h
h
is the height of the cone).
Related Rates
A plane flies 3 km over a student, moving parallel to the ground at 2km/s. How fast is its distance from the student changing when it has travelled 4 km?
Related Rates
A cylinder is being flattened so that its volume does not change. If the height of the cylinder is decreasing at 0.4 cm/s, find the rate of change of the radius when the radius is 3 cm and the height is 4 cm.
Practice: Related Rates
Practice: Related Rates
A ladder that is
13
\sqrt{13}
13
meters long is resting against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 2m/s, how fast is the top of the ladder sliding down the wall when the bottom is 3m away from the wall?
Related Rates
Suppose there is an isosceles triangle with a constant base of 10cm. If the other two equal sides are increasing at a rate of 2cm/s, at what rate is the area of the triangle increasing when each side is 10cm long?
Related Rates
A plane flies 3 km over a student, moving parallel to the ground at 2km/s. How fast is its distance from the student changing when it has travelled 4 km?
Related Rates
A
A
A
and
B
B
B
, two people of identical height
h
h
h
(in meters), stand beneath a street lamp of height
L
L
L
(in meters).
A
A
A
walks in a straight line and at a constant speed
s
s
s
(in meters per second) away from the street lamp. One second later,
B
B
B
walks in a straight line and at the same speed, but in the opposite direction, away from the street lamp. As
A
A
A
and
B
B
B
move away from the lamp, their shadows grow longer. How fast is the distance between the tips of their shadows changing 5 seconds after A started walking?
Related Rates
An ant is travelling along the path
y
=
x
2
y=x^2
y
=
x
2
and a worm is travelling along the path
y
=
∣
x
∣
y=\left|x\right|
y
=
∣
x
∣
. As the ant passes through the point
(
2
,
4
)
\left(2,4\right)
(
2
,
4
)
, the x-coordinate is increasing at a rate of 2cm/s. As the worm passes through the point
(
1
,
1
)
\left(1,1\right)
(
1
,
1
)
, the x-coordinate is increasing at a rate of 3cm/s. How fast is the distance between the ant and the worm changing at this point in time?
Related Rates
If
y
=
9
−
x
2
y=\sqrt{9-x^2}
y
=
9
−
x
2
where
x
x
x
is a function of
t
t
t
, and
d
x
d
t
=
5
\frac{dx}{dt}=5
d
t
d
x
=
5
when
x
=
5
x=\sqrt{5}
x
=
5
, find
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
5
x=\sqrt{5}
x
=
5
.
Related Rates
A bunny and a turtle start from the same place at time
t
=
0
t=0
t
=
0
seconds. The bunny heads east at
5
5
5
m/s. 10 seconds later, the turtle heads south at
25
7
\frac{25}{7}
7
25
m/s and the bunny continues on with it's path. How fast is the distance between this bunny and turtle changing at
t
=
24
t=24
t
=
24
seconds?
A person is standing 120 feet away from a model rocket that is fired straight up into the air at a rate of 20 ft/sec. At what rate is the distance between the person and the rocket increasing 2.5 seconds after liftoff and 8 seconds after liftoff? (in ft/sec)
Related Rates
The radius
r
r
r
of a pizza plate increases at a rate of
.02
.02
.02
cm/min placed in an oven. At what rate is the area of the plate increasing when
r
=
40
r=40
r
=
40
cm?
A rocket rising straight up from a level field by an observer
600
600
600
feet from the point of launch. At the moment the observer's angle of elevation is
π
4
\frac{\pi}{4}
4
π
, the angle is increasings at a rate of
.1
.1
.1
rads/minute. How fast is the rocket rising at that moment?
Practice: Related Rates
A rocket rising straight up from a level field by an observer
600
600
600
feet from the point of launch. At the moment the observer's angle of elevation is
π
4
\frac{\pi}{4}
4
π
, the angle is increasings at a rate of
0.1
0.1
0.1
rads/minute. How fast is the rocket rising at that moment?
Two cylindrical swimming pods are being filled simultaneously with water at the same rate in
m
3
/
m
i
n
m^3/min
m
3
/
min
. The smaller pool has a radius of
5
m
5 m
5
m
and the height of water is increasing at a rate of
0.5
m
/
m
i
n
0.5 m/min
0.5
m
/
min
. The larger pool has a radius of
10
m
10 m
10
m
. How fast is the height of water changing in the larger pool in m/min? (don't include units in your answer)
Two cylindrical swimming pools are being filled simultaneously with water at the same rate in
m
3
/
m
i
n
m^3/min
m
3
/
min
. The smaller pool has a radius of
5
m
5 m
5
m
and the height of water is increasing at a rate of
0.5
m
/
m
i
n
0.5 m/min
0.5
m
/
min
. The larger pool has a radius of
10
m
10 m
10
m
. How fast is the height of water changing in the larger pool when the smaller pool has a height of water at
5
m
5m
5
m
, the larger pool has a height of water at
10
m
10m
10
m
, and the radius of the smaller and larger pool are changing at a rate of
1
m
min
1\ \frac{m}{\min}
1
m
i
n
m
and
0.25
m
min
0.25\ \frac{m}{\min}
0.25
m
i
n
m
respectively?
Practice: Related Rates
A cylinder is being flattened so that its volume does not change. If the height of the cylinder is decreasing at 0.4 cm/s, find the rate of change of the radius when the radius is 3 cm and the height is 4 cm.
Practice: Related Rates 2
Practice Question: Related Rates
A cylinder is being flattened so that its volume does not change. If the height of the cylinder is decreasing at 0.4 cm/s, find the rate of change of the radius when the radius is 3 cm and the height is 4 cm.
Practice: Related Rates
Practice Question: Related Rates
A plane flies 3 km over a student, moving parallel to the ground at 2km/s. How fast is its distance from the student changing when it has travelled 4 km?
Related Rates
Car A starts heading north from an intersection at 60 km/hr at the same time that car B starts heading east from the same intersection at 100 km/hr. Two hours after leaving the intersection, how fast is the distance between the cars increasing, in km/hr? (Round your answer to two decimal places.)
Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later?
65
mi/h
Related Rates
Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later?
65
mi/h
Related Rates
The length of a rectangle is increasing at a rate of 8 cm/s and its width is increasing at a rate of 3 cm/s. When the length is 20 cm and the width is 10 cm, how fast is the area of the rectangle increasing?
140
c
m
2
/
s
cm^2/s
c
m
2
/
s
final114
A spectator observes a rocket being launched from 1 km away. The rocket lifts off vertically, rising a a speed of
0.7
0.7
0.7
km/s. When its altitude is 3 km, how fast is the distance between the rocket and the spectator changing?
Related Rates
A ladder that is
13
\sqrt{13}
13
meters long is resting against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 2m/s, how fast is the top of the ladder sliding down the wall when the bottom is 3m away from the wall?
Suppose you have a circle whose radius is increasing at a constant rate of
5
5
5
m
/
s
m/s
m
/
s
. What rate is the area changing when the area is
9
π
9\pi
9
π
m
2
m^2
m
2
?
A particle is moving along the curve
y
=
2
x
2
−
3
y=2x^2-3
y
=
2
x
2
−
3
. At the point
P
(
2
,
5
)
P(2,5)
P
(
2
,
5
)
, the
x
x
x
-coordinate is changing at the rate of 2 units per second. At this point, at what rate is the distance from the origin changing?
Example: Cone
Q:
\textbf{Q:}
Q:
A conical tank with height of
6
6
6
m and the diameter of
4
4
4
m is used upside down for storing a certain liquid. The liquid is poured in at a flow of
F
F
F
. When the height of liquid in the tank is
2
π
\dfrac{2}{\sqrt{\pi}}
π
2
m, the height is increasing at the rate of
2
2
2
m/hr. Find
F
F
F
.
Related Rates
A rocket rising straight up from a level field by an observer
600
600
600
feet from the point of launch. At the moment the observer's angle of elevation is
π
4
\frac{\pi}{4}
4
π
, the angle is increasings at a rate of
.1
.1
.1
rads/minute. How fast is the rocket rising at that moment?
Related Rates
The radius
r
r
r
of a pizza plate increases at a rate of
.02
.02
.02
cm/min placed in an oven. At what rate is the area of the plate increasing when
r
=
40
r=40
r
=
40
cm?
A source of light is 10 meters above the ground. An point object is falling vertically and landing 4 meters away from the base of the light. When the object is below the source of light, the light casts a shadow on the ground. When the object is 4 meters above the ground (with speed of 10 meters per second), how fast is the shadow moving? How fast is the distance between the object and its shadow changing?
Practice: Similar Triangles
Q.
\textbf{Q.}
Q.
A source of light is at a height of 10 meters above the ground. An object is falling vertically at a speed of 10 meters per second and landing 4 meters away from the base of the light. When the object is below the source of light, 5 meters above the ground, the light casts a shadow on the ground. How fast is the shadow moving? How fast is the distance between the object and its shadow changing?
final114
A spectator observes a rocket being launched from 1 km away. The rocket lifts off vertically, rising a a speed of
0.7
0.7
0.7
km/s. When its altitude is 3 km, how fast is the distance between the rocket and the spectator changing?
Circle Equation
An object is moving on a circular path around the origin with a radius of
13
13
13
m. If its image on the
x
−
x-
x
−
axis moves at the speed of
3
3
3
m/s, how fast is its image on the
y
−
y-
y
−
axis moving when
x
=
5
x = 5
x
=
5
m?
Triangle with Pythagorean Theorem and Trigonometry
A plane is located at point
(
3
,
4
)
(3,4)
(
3
,
4
)
from the origin (distances are measured in km). It moves towards the right at a speed of
1
1
1
km/s. How fast is its distance from the origin changing at that time? How fast is the angle of elevation changing at that time?
Related Rates: Triangle with Trigonometry
A rocket rising straight up from a level field is observed by an observer
600
600
600
ft from the point of launch. At the moment the observer's angle of elevation is
π
/
4
\pi/4
π
/4
, the angle is increasing at the rate of
0.1
0.1
0.1
rad/min. How fast is the rocket rising at that moment?
Practice: Rate of Change of Coordinate
Q:
\textbf{Q:}
Q:
A point moves on the graph of
y
=
x
+
2
x
−
1
\displaystyle y=\frac{x+2}{x-1}
y
=
x
−
1
x
+
2
. When
x
=
2
x=2
x
=
2
the rate of change along the x-axis is 4 units per second. What is the rate of change along the y-axis?
A man running straight at a rate of
10
10
10
ft/sec along a level street passes under a vertically rising balloon when it is
50
50
50
ft high. Assuming that the balloon is rising at a constant rate of
1.5
1.5
1.5
ft/sec, how fast is the distance between the runner and the balloon increasing
2
2
2
seconds later?
Practice: Sphere
Q:
\textbf{Q:}
Q:
The volume of a spherical snow ball is decreasing at the rate of
50
π
50\pi
50
π
ft
3
^3
3
/min. What is the rate of change of the radius when
r
=
3
r=3
r
=
3
ft?
Related Rates: Sphere
The volume of a spherical snow ball is decreasing at the rate of
50
50
50
ft
3
/min. What is the rate of change of the area when
r
=
3
r=3
r
=
3
ft? Is the area increasing or decreasing at this instant?
Two planes take off from an airport at the same time. One is heading east at
100
km
h
100\,\frac{\text{km}}{\text{h}}
100
h
km
and the other one is heading south-east at
200
2
km
h
200\sqrt2\,\frac{\text{km}}{\text{h}}
200
2
h
km
.At what rate is the distance between them is changing after one hour?
Related Rates
At 1 pm a ship A is 25 km due north of ship B. If ship A is sailing west at a rate of 16 km/h and ship B south at 20 km/h, dinf the rate at which the distance between the ships is changing at 1:30 PM
A spherical ball of ice melts so that its radius decreases from 10 cm to 8 cm. By approximately how much (in cubic centimetres) does the volume of the ball change? Include the negative sign in your answer if the volume is decreasing.
A
10
f
t
10 ft
10
f
t
ladder is resting against a wall with the base of the ladder initially
5
f
t
5 ft
5
f
t
from the wall. If the bottom gets pushed towards the wall at a constant rate of
0.5
f
t
/
s
0.5 ft/s
0.5
f
t
/
s
, how fast is the top of the ladder moving up the wall after 8 seconds?
A spectator observes a rocket being launched from 1
k
m
km
k
m
away. The rocket lifts off vertically, rising a a speed of
0.7
0.7
0.7
k
m
/
s
e
c
km/sec
k
m
/
sec
. When its altitude is 3
k
m
km
k
m
, how fast is the distance between the rocket and the spectator changing?
Suppose you have a circle whose radius is increasing at a constant rate of
5
5
5
m
/
s
m/s
m
/
s
. What rate is the area changing when the area is
9
π
9\pi
9
π
m
2
m^2
m
2
?
Related Rates
The volume of a spherical snow ball is decreasing at a rate of
50
π
f
t
3
/
m
i
n
50\pi ft^3/min
50
π
f
t
3
/
min
. What is the rate of change of the radius when
r
=
3
r=3
r
=
3
ft?
Related Rates
A man running straight at a rate of
10
10
10
ft/sec along a level street passes under a vertically rising balloon when it is
50
50
50
feet high. Assuming that the balloon is rising at a constant rate of
1.5
1.5
1.5
ft/sec, how fast is the distance between the balloon and the running increasing
2
2
2
seconds later?
Practice: Related Rates
Q:
\textbf{Q:}
Q:
You are marching in a parade down Government St. Your parade floats move in a straight line at a speed of 2 m/s. Your parade float has a disco ball mounted to it that moves up and down. The height of the disco ball in meters is given by the equation
h
(
x
)
=
3
sin
(
π
10
x
)
+
4
h(x)=3\sin\left(\frac{\pi}{10}x\right)+4
h
(
x
)
=
3
sin
(
10
π
x
)
+
4
where
x
x
x
is meters from the start of the Government St. At what rate is the disco ball gaining height 70 seconds into the parade (i.e., what is the vertical component of its velocity)?
Practice: Related Rates
A cylinder is being flattened so that its volume does not change. If the height of the cylinder is decreasing at 0.4 cm/s, find the rate of change of the radius when the radius is 3 cm and the height is 4 cm.
A spherical balloon is filled with air. If its radius increases at 2 cm/s, how fast is its volume changing when 𝑟 = 5 𝑐𝑚?
Related rates
A snowball is melting such that its volume is decreasing at a rate of
2
c
m
3
s
2\ \frac{cm^3}{s}
2
s
c
m
3
. Find the rate at which the diameter is changing when the surface area is
16
π
c
m
2
16\pi\ cm^2
16
π
c
m
2
.
Related Rates
Suppose there is an isosceles triangle with a constant base of 10cm. If the other two equal sides are increasing at a rate of 2cm/s, at what rate is the area of the triangle increasing when each side is 10cm long?
Related Rates
A cylinder is being flattened so that its volume does not change. If the height of the cylinder is decreasing at 0.4 cm/s, find the rate of change of the radius when the radius is 3 cm and the height is 4 cm.
Related Rates
A plane flies 3 km over a student, moving parallel to the ground at 2km/s. How fast is its distance from the student changing when it has travelled 4 km?
Practice: Related Rates
Practice: Related Rates
A ladder that is
13
\sqrt{13}
13
meters long is resting against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 2m/s, how fast is the top of the ladder sliding down the wall when the bottom is 3m away from the wall?
A person is pulling a weight attached to a rope of fixed length through a hook in the ceiling as in the diagram below. Directly below the hook is point
A
A
A
on the ground. The rope is fully stretched with the ceiling
3
3
3
meters above the ground. The weight is being dragged along the ground by the person pulling on the rope. When the person is
3
3
+
3
3 \sqrt{3} + 3
3
3
+
3
meters away, the weight is
3
3
3
meters away from
A
A
A
, and the person is traveling at
1
1
1
m/s, how fast is the weight travelling? (Diagram is not to scale)
A person is pulling a weight attached to a rope of fixed length through a hook in the ceiling as in the diagram below. Directly below the hook is point
A
A
A
on the ground. The rope is fully stretched with the ceiling
3
3
3
meters above the ground. The weight is being dragged along the ground by the person pulling on the rope. When the person is
3
3
+
3
3 \sqrt{3} + 3
3
3
+
3
meters away, the weight is
3
3
3
meters away from
A
A
A
, and the person is traveling at
1
1
1
m/s, how fast is the weight travelling? (Diagram is not to scale)
Related Rates
An inverted conical tank of water is used as a water trough to hydrate some animals on a farm. However, there is a leak at the bottom of the tip causing water to drip out at a rate of
1000
c
m
3
/
s
1000 \; cm^3 /s
1000
c
m
3
/
s
. The height of the trough is
5
m
5 \; m
5
m
and the diameter at the top of the trough is
4
m
4 \; m
4
m
. (The volume of a cone is
1
3
π
r
2
h
\frac{1}{3} \pi r^2 h
3
1
π
r
2
h
where
r
r
r
is the radius of the base and
h
h
h
is the height of the cone).
A person is standing 120 feet away from a model rocket that is fired straight up into the air at a rate of 20 ft/sec. At what rate is the distance between the person and the rocket increasing.
Related Rates
Paul asks his friend to record himself skateboarding off a ramp. The ramp has a slope of
1
2
\frac{1}{2}
2
1
while the ramp itself is 2 meters high. Paul's friend is standing 4 meters away on a platform so that it is level with the top of the ramp with the camera pointed at the ramp. The angular elevation of the camera,
θ
(
t
)
\theta(t)
θ
(
t
)
, starts at 0 and increases with time so that the camera follows the trajectory of Paul. When Paul is at a horizontal distance of
x
x
x
from the ramp, the vertical distance he has is measured by
f
(
x
)
f(x)
f
(
x
)
from the top of the ramp.
Assuming the Paul is moving horizontally at a speed of 5 m/s when he jumps off the ramp, how quickly is his friend rotating the camera in order to follow his trajectory as he leaves the ramp? Note that
f
(
0
)
=
0
f(0) = 0
f
(
0
)
=
0
and that
f
′
(
0
)
=
1
3
f'(0) = \frac{1}{3}
f
′
(
0
)
=
3
1
.
(Diagram not to scale)
Practice: Related Rates
A man running straight at a rate of
10
10
10
ft/sec along a level street passes under a vertically rising balloon when it is
50
50
50
feet high. Assuming that the balloon is rising at a constant rate of
1.5
1.5
1.5
ft/sec, how fast is the distance between the balloon and the running increasing
2
2
2
seconds later?
Practice: Related Rates
Q:
\textbf{Q:}
Q:
You are marching in a parade down Government St. Your parade floats move in a straight line at a speed of 2 m/s. Your parade float has a disco ball mounted to it that moves up and down. The height of the disco ball in meters is given by the equation
h
(
x
)
=
3
sin
(
π
10
x
)
+
4
h(x)=3\sin\left(\frac{\pi}{10}x\right)+4
h
(
x
)
=
3
sin
(
10
π
x
)
+
4
where
x
x
x
is meters from the start of the Government St. At what rate is the disco ball gaining height 70 seconds into the parade (i.e., what is the vertical component of its velocity)?
Practice: Related Rates
The volume of a spherical snow ball is decreasing at a rate of
50
π
f
t
3
/
m
i
n
50\pi ft^3/min
50
π
f
t
3
/
min
. What is the rate of change of the radius when
r
=
3
r=3
r
=
3
ft?
Two cylindrical swimming pods are being filled simultaneously with water at the same rate in
m
3
/
m
i
n
m^3/min
m
3
/
min
. The smaller pool has a radius of
5
m
5 m
5
m
and the height of water is increasing at a rate of
0.5
m
/
m
i
n
0.5 m/min
0.5
m
/
min
. The larger pool has a radius of
10
m
10 m
10
m
. How fast is the height of water changing in the larger pool?
Related Rates
An object travels on a path that draws out a square which is 20 meters on a side. The object starts at corner A, and moves towards corner B. When it is halfway to corner B with a speed of 10 meters per second, at what rate is the object’s distance from corner C (the corner directly opposite corner A) changing? Do not simplify your answer.
Related Rates
A source of light is 10 meters above the ground. An object is falling vertically and landing 4 meters away from the base of the light. When the object is below the source of light, the light casts a shadow on the ground. When the object is 4 meters above the ground (with speed of 10 meters per second), how fast is the shadow moving?How fast is the distance between the object and its shadow changing?
Practice: Related Rates
If a tree trunk adds
1
/
4
1/4
1/4
of an feet to its diameter and
1
1
1
foot to its height each year, how rapidly is its volume changing when its diameter is
3
3
3
feet and its height is
50
50
50
feet (assume that the tree trunk is a circular cylinder).
Practice: Related Rates
A rocket rising straight up from a level field by an observer
600
600
600
feet from the point of launch. At the moment the observer's angle of elevation is
π
4
\frac{\pi}{4}
4
π
, the angle is increasings at a rate of
.1
.1
.1
rads/minute. How fast is the rocket rising at that moment?