The ladder and box problem,
This is one of the so-called "classic" problems.
A box with width A and height B stands next to a wall. You put a ladder of length C next to the wall over the box as shown in the picture below.
At what height H does the ladder touch the wall?
It is a nice math problem because a solution can be obtained in both an easy and a hard way, depending on the approach that is taken.
Lesson plan
Students work in small groups. Each group is given a box or a wooden block, and a stick (or a chopstick or a straw) representing a ladder, and a scientific calculator, TI-84. They may also need a little bit of clay or play dough to put on the end of the stick so that it will not slide.
The task is:
(1) Find a general way of solving this problem for any values A, B, and C.
(2) Measure the width and A and length B of the box, and the length C of the stick, and compute H.
(3) Put the box next to a wall or any other vertical surface, put the stick next to it and measure how high up it touches the wall.
(4) Compare the predicted and actual results, and discuss the whole problem.
The teacher should let students present or propose solutions. But if none of their solutions is as simple as the one shown below, then the solution below should be implemented.
A solution
Let X be the angle between the ladder and the ground, and C1 and C2 be the lengths of parts of the ladder below and above the top of the box.
So we have:
C1*sin(X) = B
C2*cos(X) = A
C1 + C2 = C
By replacing C1 and C2 we get,
B/sin(X) + A/cos(X) = C
We can solve this equation for X using SOLVER, and then we get
H = C*sin(X)
Warning: There are two solutions. The ladder can reach the wall either high or low.
Example
Let A = 2 in., B = 3 in., C = 11 in.
Define: Y1=B/sin(X)+A/cos(X) - C
Enter: 2→A:3 →B:11 →C ENTER
Set the MODE to DEGREE, WINDOW to Xmin=0, Xmax=90, Xscl=10, Ymin=-11,
Ymax=1, and GRAPH in order to see both solutions.
(You see that the bigger angle X is between 70 and 80 degrees)
In SOLVER put
Y1=0
X=65
bound={-1E99,1...
and ALPHA SOLVE
You get
X=75.333154185... (You do not need such precision!)
From the home screen
Csin(X) ENTER
10.64155871
which is approximately 10 5/8 inches.
To find the second solution, in SOLVER put
X=30.
You get
X=19.757003605
Then from the home screen,
Csin(X)
3.718349371
which is approximately 3 23/32 inches.