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Demonstration Station - Shadows

Calc12 - Related Rates Stations Activity Name: __________________

Time	Distance from wall	Height of shadow	Δdistance	Δheight
0	0 cm	cm	0	0
1
2
3
4
5

Related rates problems in calculus are an opportunity to experiment and make observations. We can then use these real-world observations to help understand the mathematics that explain the relationships we observe.

These stations are designed to help you see:

How a change in one variable affects the change in another related variable in the real-world by making observations and collecting data.
How our calculus formulas match the data and the observed rates of change.

You will work in small groups to make observations at a number of stations involving related rates. Each student will turn in their own packet.

To start, we will make observation of the first station as a whole class.

Diagrams and Equations - Draw a diagram and write an equation that describes the relationship between the variables in this station. Then state what the derivative of this equation would be with respect to time.

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Instructions: Complete the following stations in any order.

Station 1 - Blowing Up!

Breaths	Circumference	Radius	Volume	Δradius	Δvolume
0	0 cm	0 cm	0 cm³	0	0
1
2
3
4
5

Station 2 - Slip Sliding Away

Time	distance of base from the bottom of the wall	Height the ladder reaches on the wall	Δbase	Δheight
0	0 cm	100 cm	0	0
1
2
3
4
5
6
7
8
9
10	100 cm	0 cm

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Station 3 - Don’t Fence Me In

Station 4 - You Got a Fast Car

Rectangle	Length	width	area	Δlength	Δwidth	Δarea
1 (4x2)				0	0	0
2 (6x3)
3 (8x4)
4 (10x5)
5 (12x6)

Time	A distance from start	B distance from start	distance between cars	ΔA	ΔB	Δdistance
0	0 cm	0 cm	0 cm	0	0	0
1
2
3
4
5

Reflection - How did you make sure you were collecting accurate data? Were you surprised by the results of the data for each station, why or why not?

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Equations - Draw a diagram and write an equation that describes the relationship between the variables for each station (be sure to define your variables). Then state what the derivative of this equation would be with respect to time.

Station 1	Station 2
Station 3	Station 4

Finish off with a “does calculus really work” discussion. Take the sliding ladder station as an example. Look at time 6 when the ladder is 60 cm from the bottom of the wall. Using Pythagorean Theorem we can find that the “ladder” should be 80 cm on the wall (6-8-10 Pythagorean triple). What was your observed Δheight? Does this match the calculated dh/dt for this time (with b=60, h=80, db/dt = 10)? Why does it not match exactly? How could we get more detailed observations? Lets try it…

Station #1

V = (4/3)𝜋r³

dV/dt = 4𝜋r²(dr/dt)

dr/dt = (dV/dt)/4𝜋r²

Station #2

b= distance from wall to bottom of ladder

h= height of ladder up the wall

l= length of the ladder = 100cm

db/dt = 10 cm/time

b² + h² = l²

b² + h² = 100²

2b(db/dt) + 2h(dh/dt) = 0

2b(10) = -2h(dh/dt)

dh/dt = -(20b)/2h

dh/dt = -(10b)/h

At time 6, dh/dt = -(10*60)/80 = -7.5cm/time

Station #3

A = area, l = length, w = width

A = lw

dA/dt = (dl/dt)w + (dw/dt)l

At time 4, dA/dt = (2)(5) + (1)(10)

dA/dt = 20 units^2/time

Station #4

a= distance from start of car A

b= distance from start of car B

c= distance between the two cars

da/dt = 8cm/time, db/dt= 12cm/time, dc/dt = ?

a² + b² = c²

2a(da/dt) + 2b(db/dt) = 2c(dc/dt)

2a(8) + 2b(12) = 2c(dc/dt)

dc/dt = [2a(8) + 2b(12)] / 2c

dc/dt = (8a + 12b) / c