Modeling a Ladder Problem Name(s):

Drawing diagrams is a useful method to help solve many types of realisticproblems. Dynamic diagrams can be even more useful. F{ere's a problemthat can be solved with a Sketchpad sketch.

The Occupational Safety and Health Administration (OSHA) recommendsthat when you use a ladder, you should lean it against a wall so that theheight at which it touches the wall is four times the distance from thewall to the foot of the ladder. Any more and you risk tipping the ladderbackward. A^y less and you risk having the bottom slide out from underthe ladder. What's the height from the floor that you can reach with a

20-foot ladder? What angle will the ladder make with the floor?

choose I Sketch and lnvestigatePreferences from ,

the Edit menu I -':;o'il il:ll: it 1. Set Preferences to display the DistanceUnits panel I Units in inches.

,-=1.TSl,::_i::,,::"jl horizontal segmentAC. Thesesegments- .".t"i.;;;;t:. I represent the wall and the floor.

3. Construct point D on the floor. This pointwill be the foot of your ladder.

:e :;t point D; then,^ the Transform

menu, chooseTranslate.

Translate point D vertically by 2 inches.The 2 inches will represent the length of yourladder, so the scale of your drawing will be L in.

Construct circle DD'.

Construct point E where the circle intersects the wall. You may haveto move point D first so that the circle and the wall intersect.

Construct pJ. tnis segment represents your ladder. Its length can'tchange because the radius of the circle is fixed at 2 inches.

Hide the circle and point D'.

Drag point D back and forth. You should see the top of the laddermove up and down the wall.

Measure IEDA, EA, and AD. (EA represents the height on the wallthat your ladder is reaching.) Calculate EA/ AD.

Drag point D. Given the constraints in the problem, how high can theladder reach? What angle does it make with the fioor?

5.

6.

7.

B.

9.

i= :ct, in order, Ir: -'. E, D, and B. I--r' ^ :1e Measure f>]Q.

-:nu. choose I

flb 3elect points I

: =" - -: then, in the I

" ,,-.".1?:?$T:'l 01:=:eat for AD. I

l&Milrrr:-: Geometry with The Geometer's SketchpadI .1;yl Cuniculum Press

Chapter 10; Trigonometry and Fratia s . 197

Modeling a Ladder Problem (continued)

Q2 Confirm your answers using trigonometry. Show your work.

Explore More

1. Suppose a ladder is propped against one wall in the corner of a room.To one side of the ladder is another wall. A wet paintbrush rests onthe center rung of the ladder, just touching the side wall. Suddenly,the foot of the ladder slips and the paintbrush falls with it, paintinga skeak on the side wall as it falls! What does the streak look like?To model this in your sketch, construct the midpoint of your ladder.\Alhile it's selected, choose Trace Point in the Display menu. Animatepoint D along BZ

2. Select the measurements Ior EA and AD and choose Plot As (x, y) inthe Graph menu. Drag the foot of the ladder. What kind of graph doyou get? If you were to drag the foot of a ladder away from a wall at a

constant rate, would the top of the ladder fall at a constant rate? Whyor why not?

3. Write one or more other problems that could be modeled with thissketch.

198 . Chapter 10: Trigonometry and Fractals Exploring Geometry with The Geometer's Sketchpad

@ 2002 Key Curriculum Press

A Sine Wave Tracer Name(s):

In this exploration, you'll construct an animation "engine" that traces outa special curve called a sine Ttiare. Variations of sine curves are the gaph-.of functions called periodic functions, functions that repeat themseh-es. Themotion of a pendulum and ocean tides are examples of periodic functiorL-s.

Sketch and lnvestigate

1. Construct a horizontal segment AB.

2. Construct a circle with center A and radius endpoint C.

Construct point D onAB.

Construct a line perpendicular to AA tfuough point D.

Construct point E on the circle.

Construct a line parallel to AB through point E.

Construct point F, the point of intersection of the vertical line throughpoint D and the horizontal line through point E.

Drag point D and describe what happens to point F.

J.

Select point Dand AB: then, in

the Constructmenu, choose

PerpendicularLine.

4.

5.

6.

7.

"'lY,?J'/;lI:Jil'i I'a{

Q2 Drag point E around the circle and describe what point F does.

Q3 In a minute, you'll create an animation in your sketch that combinesthese two motions. But first try to guess what the path of point F willbe when point D moves to the right along the segment at the sametime that point E is moving around the circle. Sketch the path youimagine below.

&ffircrrg Geometry with The Geometer's SketchpadO mi2 Key Curriculum Press

Chapter 10: Trigonometry and Fractals . 199

n 9tI I9 tt qYg I I qvEr \vvt trtr rsgv]ir

.t:iT:i:[:3"111 lt 8. Make an action bution that animates point D fonr-ard along.r5 a'-:Action Buttons I I point E forward around the circle.

Animation. I

Choose forward I -" ;","h-"';iJ"il; | 9. Move point D so that it's just to the right of the circle.pop-up menu for

I

point D I 10. Select point F; then, in the Display menu, choose Trace Point.

11. Press the Animation button.

Q4 In the space below, sketch the path traced by point F. Does the act:.-path resemble your guess in Q3? How is it different?

12. Select the circle; then, in the Graph menu, choose Define Unit CircleYou should get a graph with the origin at pointA Point B should lieon the x-axis. The y-coordinate of point F above AB is the value of th.-

sine of IEAD.

A5 If the circle has a radius of 1 grid unit, what is its circumference ingrid units? (Calculate this yourself; don't use Sketchpad to measureit because Sketchpad wiil measure in inches or centimeters, not gridunits.)

13. Measure the coordinates of point B.

14. Adjust the segment and the circle until you can make the curvetrace back on itself instead of drawing a new curve every time.(Keep point B on the r-axis.)

QG What's the relationship between the r-coordinate of point B and thecircumference of the circle (in grid units)? Explain why you think thisis so.

E F

cD B

A 105

200 . Chapter 10: Trigonometry and Fractals Exploring Geometry with The Geometer's Sketchpad

@ 2002 Key Curriculum Press