Building a Skeletal Box according to Specifications
You will need a TI-83/84 calculator for this unit!
You have a stick that is one yard (36 inches) long. Your task is to use it to build a skeletal rectangular solid that has volume 24 cubic inches and surface area 52 square inches. The edge lengths of the skeleton should sum to 36 inches (you should use the whole stick).
Plan first before you cut your stick!
Part 1. Algebraic solution.
Let the length, width, and height of the rectangular solid be a, b, and c.
We know:
4(a + b + c) = 36 inches (Why?)
So a + b + c = 9 inches (1)
a*b*c=24 cubic in. (2)
2(a*b + a*c + b*c) = 52 square in. (Why?) (3)
(So a*b + a*c + b*c = 26 square in.)
(1), (2), and (3) above are the three specifications.
What next?
Consider the equation
(x-a)(x-b)(x-c)=0
If we multiply it out, we will find something that will be helpful.
(x − a) * (x − b) * (x − c) = 0
(x 2− (a + b)x + ab) * (x − c) = 0
x 3− (a + b)x 2+ abx − cx 2+ c(a + b)x − abc = 0
x 3− (a + b + c)x 2+ (ab + ac + bc)x − abc = 0
Look at the last three coefficients:
a + b + c,
ab + ac + bc, and
abc.
Do you see that
a * b * c = 24 cu. inches = volume
So we can write
x 3− (a + b + c)x 2+ (ab + ac + bc)x − abc = 0
as
x 3− 9x 2+ 26x − 24 = 0
Now we have a cubic polynomial equation in just one variable.
And if we solve it, (if there are solutions), we will find a, b, and c:
(x-a)(x-b)(x-c)=0
We will use Solver on the TI-83/84.
Under Y=, enter
Y1=X3-9X2+26X-24
Now go to Solver (under Math).
You will see
EQUATION SOLVER
eqn:0=
Enter
eqn:0=Y1
Press ENTER, and you will see
Y1=0
X=(something)
bound={-1E99,1E99}
Enter a guess at X=, and then press
ALPHA SOLVE.
If you made a good guess, you will see one solution.
Now make a second guess, and if it is a good one, you will get a second solution.
Finally, with a third good guess, you will get a third solution.
Now check if the three solutions you found, call them a', b', and c', yield
a'*b'*c'= 24
a'+b'+c'=9
and
a'*b' + a'*c'+b'*c'=26.
If they are correct, you may cut your stick.
Part 2. Building the skeleton
This is not so easy! We found 36-inch long thin sticks at Hobby Lobby, and we used an X-Acto knife to cut it. You need to cut the stick into 12 pieces, 4 with length a, 4 with length b, and 4 with length c.
How to make the 8 vertices?
We found that wood glue worked fairly well.
We worked with gluing two edges at a time. But you need to wait a while for each intersection to dry. Here is a picture of my finished product:
