CALCULUS LAB: ANIMATIONS AND RELATED RATES
PART ONE: THE SLIDING LADDER

Open the related rates geometer’s sketchpad file. You should see the following
problem description:
A 6-m. ladder is against a wall. If its bottom
is pulled/pushed at a constant 1/2 m/sec,
how fast is the ladder top sliding when it
reaches 5m, 3m,1m up the wall?
Let’s focus on the question of determining the rate at various points along the wall
as we pull the ladder away from the wall.

Click on the “Slide Ladder” button and watch the ladder as it falls down the wall.
Does it seem to be moving down the wall at a constant speed, an increasing
speed, or a decreasing speed?

Click on the “Show dy/dt” button.
On the time interval from t = 0 to t = 12, what is dy/dt representing? Why is
dy/dt negative?

Click on the “Y=5” button and record the value for dy/dt. (GSP uses D y t to
represent dy/dt.) We will be using calculus to confirm this value.

Click on the “Y=1” button and record the value for dy/dt. We will also be
confirming this value.

On to the calculus. Hooray!!!!
USING CALCULUS TO CONFIRM THE LADDER ANIMATION
a) Draw a sketch of the situation described above. A side-view of a ladder propped
up against a wall would be best. Label the sketch with variables for quantities
that are changing with respect to time. Label constants for only those quantities
that remain constant at all times t.
b) Identify the known rate information in symbols. Identify the unknown rate in
symbols. Then, state the specific goal. In other words, what is it, in symbols,
that we are trying to confirm?
c) Write an equation that defines a known relationship between the original
quantities in your diagram (not the rates).
d) Differentiate your equation (both sides) with respect to time.
e) Find the specific solutions (when Y=1 and when Y=5) so that we can confirm the
values we recorded in the animation.
PART TWO: REELING IN A BOAT

Click on the tab at the bottom of the screen to switch to the animation related to
the boat and the winch (reel). You should see the following problem description:
A winch (alt. 20 ft) reels in a rope at
2 ft/sec. How fast is the boat moving
when the rope = 45, 30, 22, 20.05 ft?

Click on the “Reel In” button and watch the boat as it is reeled in toward the
dock.
Does the boat seem to be moving toward the dock at a constant speed, an
increasing speed, or a decreasing speed?

Click on the “Show dx/dt” button.
What is the graph of dx/dt representing? Why is dx/dt negative?

Click on the “Rope = 45” button and record the value for dx/dt. We will be using
calculus to confirm this value.

Click on the “Rope = 20.05” button and record the value for dx/dt. We will also
be confirming this value.

On to the calculus again. Hooray!!!!
USING CALCULUS TO CONFIRM THE BOAT ANIMATION
a) Draw a sketch of the situation described above. Label the sketch with variables
for quantities that are changing with respect to time. Label constants for only
those quantities that remain constant at all times t.
b) Identify the known rate information in symbols (be careful with the sign).
Identify the unknown rate in symbols. Then, state the specific goal. In other
words, in symbols, what is it that we are trying to confirm?
c) Write an equation that defines a known relationship between the original
quantities in your diagram (not the rates).
d) Differentiate your equation (both sides) with respect to time.
e) Find the specific solutions (for r = 40 and r = 20.05) so that we can confirm the
values we recorded in the animation.
PART THREE: SEARCHLIGHT

Click on the tab at the bottom of the screen to switch to the animation related to
the searchlight. You should see the following problem description:
A plane (alt. 4000 ft) is flying west at 700 ft/s. A searchlight, under its path,
tracks it. How fast is the light pivoting when the plane is 1000 ft east,
overhead, 5000 ft west?

Click on the “Fly” button and watch the plane fly from the east to the west.
Does the angle between the searchlight and the vertical seem to be changing
at a constant rate, an increasing rate or a decreasing rate (in magnitude) as
the plane approaches the vertical?

Click on the “Show
d
” button.
dt
What is the graph of

d
representing?
dt
Why is the graph of
d
negative and decreasing for the first half?
dt
Why is the graph of
d
positive and decreasing for the second half?
dt
Click on the “1000 feet East” button and record the value for
using calculus to confirm this value.

On to the calculus again. Hooray!!!!
d
. We will be
dt
USING CALCULUS TO CONFIRM THE SEARCHLIGHT ANIMATION
a) Draw a sketch of the situation described above for a plane that is on the East
side of the search light. Label the sketch with variables for quantities that are
changing with respect to time. Label constants for only those quantities that
remain constant at all times t.
b) Identify the known rate information in symbols (be careful with the sign).
Identify the unknown rate in symbols. Then, state the specific goal. In other
words, in symbols, what is it that we are trying to confirm?
c) Write an equation that defines a known relationship between the original
quantities in your diagram (not the rates).
d) Differentiate your equation (both sides) with respect to time.
e) Find the specific solution (for a plane 1000 east of the searchlight) so that we can
confirm the values we recorded in the animation.
PART FOUR: LAMP POST

Click on the tab at the bottom of the screen to switch to the animation related to
the lamp post. You should see the following problem description:
A man (6 ft) walks away from a
lamp post (15 ft) at 5 ft/sec.
How fast is his shadow lengthening?
How fast is the shadow's tip moving?

Click on the “walk” button and watch the person and the shadow move.
Does the length of the shadow seem to be increasing at a constant rate, an
increasing rate or a decreasing rate (in magnitude) as the person walks?
Does the tip of the shadow seem to be increasing at a constant rate, an
increasing rate or a decreasing rate (in magnitude) as the person walks?

Click on the “Show Rates” button.
What is the constant rate at which the shadow is lengthening?
What is the constant rate at which the tip of the shadow is moving?

On to the calculus again to confirm these constant rates. Hooray!!!!
USING CALCULUS TO CONFIRM THE LAMP POST ANIMATION
f) Draw a sketch of the situation described above. Label the sketch with variables
for quantities that are changing with respect to time. Label constants for only
those quantities that remain constant at all times t.
g) In addressing the first question about the shadow length, identify the known
rate information in symbols. Identify the unknown rate in symbols. Then, state
the specific goal. In other words, what is it, in symbols, that we are trying to
confirm?
h) Write an equation that defines a known relationship between the original
quantities in your diagram (not the rates). Hint: You’ll need similarity.
i) Differentiate your equation (both sides) with respect to time. Does this confirm
the value we recorded in the animation for the rate at which the shadow is
lengthening?
j) Now, if we know how fast the person is walking and how fast the shadow is
lengthening, how can we quickly confirm the rate at which the shadow tip is
moving? (If not obvious, you can always use calculus!)