You are given a cube. Design and construct a pyramid with square base that fits over the cube, touching its four top edges. The pyramid should have volume as small as possible.

We define a new independent variable t as follows:

But notice, because of similar triangles, that

The volume V of a pyramid is 1/3*(area of base)*height. Our goal is to make V as small as possible for our given a.

We can write the volume in terms of a and t:

Method 1: Using nDeriv

First we measure the edge length a of our cube. We store this value in a. Here we used cubic inch blocks, so

1→A

In the above formula for V, we use X for the independent variable (it is easier to use than T).

Enter

\Y

2)

Go to MATH and then select SOLVER. Press the up-arrow, and you will see

EQUATION SOLVER

eqn:0=

Enter

eqn:0=nDeriv(Y

You will see

nDeriv(Y

A=1

X= (something)

bound={-1E99,1E99}

Set the bound to

bound={0,1E99}

X is the ratio of the height of the pyramid to half of its base. So what is a good guess for this ratio? Let's try 2.

X=2

Leave the cursor on the X=2 line, and press ALPHA SOLVE.

You will see

X=4.0000001388215

So h/(b/2) = 4, and thus

h = 2*b

So the height of the pyramid is twice the base.

What is the volume of the pyramid? On the home screen, enter Y

Y

So the ratio of the volume of the pyramid to the volume of the cube is 2.25. (This ratio will hold for any cube!) The cube fills .444 of the pyramid, a little less than half.

Set WINDOW:

Xmin=0

Xmax=12

Xscl=1

Ymin=0

Ymax=6

Yscl=1

Xres=1

Then graph Y

We want to find where the Y-value of this graph is at a minimum.

2nd CALC

minimum

You need to choose a left bound and a right bound, and guess a minimum. Then press ENTER.

Minimum

X=4.0000004 Y=2.25

{.25,.5,.75,1,1.25,1.5,1.75,2,2.25,2.5,2.75,3,3.25,3.5,3.75,4,4.25,4.5,4.75,5,5.25,5.5,5.75,6}→X

"4/3*A

"ΔList(

R is a list of rates of change, namely, the difference of two adjacent volumes divided by the difference of two adjacent ratios of height to one-half of base (our variable X).

Now in STAT Edit, you may first clear out old lists by pressing DEL. Then enter:

Name=X

Name=V

Name=R.

The three lists will fill. Investigate where R is close to zero.

You will see that when X=4, V=2.25.

You can put in new values for X, slightly less than and greater than 4, to investigate this.

Is this cool, or what??

Okay, now we have to figure out how to design our pyramid!

Our pyramid will have a square bottom face, and 4 lateral faces that are congruent isosceles triangles.

1. Compute the edge length b of the base.

We know X (originally our t) is 4, and a is 1 inch.

Here is my net:

Calculus Index