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# Maximizing Triangle Area

Following is the problem we worked on early this year in Precalculus:

Take a sheet of paper and fold the

upper left corner so that it touches

some point on the bottom edge, as

illustrated here. Consider the area of

the triangle labeled “A” – the one

formed in the lower left corner of the

paper. Find the dimensions of the

triangle that has the largest area.

Students worked in groups to collect data; each group had different-sized sheet
of paper.

Expectations included a written report that included both a “data solution” and
an “algebraic

solution.” This talk looks at those solutions, plus extensions involving more
data analysis and

calculus.

Let’s begin with a sample data set collected from folding an 8.5” x 11” sheet of
paper:

And some plots of our data:

**Area (cm^2) vs. Horiz Leg (cm)**

**Area (cm^2)**

By inspecting our data and scatterplot, we see the maximum area is approximately
44.4 cm^2,

when the dimensions are approximately 12 cm x 7.4 cm.

What functions best describe the relationships in the scatterplots above?
Quadratic? Some other

polynomial? Something else? (Will revisit this question soon…)

Or, can we predict the vertical leg length if we know the horizontal leg length?

**Vert Leg (cm) vs. Horiz Leg (cm)**

Seems to be parabolic? (The quadratic fit on the next page seems promising – the
residuals show

no obvious pattern!)

**Bivariate Fit of Vert Leg (cm) By Horiz Leg (cm)
Polynomial Fit Degree=2**

Vert Leg (cm) = 10.7646 - 0.000571 Horiz Leg (cm) - 0.023314 (Horiz Leg (cm))^2

So, we could write , and with , we

have . (Where and are the vertical

and horizontal legs of the triangle, respectively.) Then if we graph this function, we notice it

matches our “area vs horiz. leg” scatterplot well and we can use our calculator to find the

maximum area of 44.56 cm^2, with a horiz. leg length of 12.4 cm. Then (12.4)= 7.18 cm

would be the vertical leg length.

What about domain?

Notice that our model A(h) (as given above) is a cubic polynomial. Polynomials have domains

of all real numbers, but is this a sensible domain in the context of our problem? It would only

make sense to have A(h)> 0, and a quick inspection reveals that for this to be the case, we must

have 0 <h< 21.5 (approximately). (Why 21.5? Notice this is approximately the shorter

dimension of the sheet of paper.)

Algebraic Approach

Realizing that the hypotenuse of our right triangle is simply the difference
between the shorter

side of the sheet of paper and the triangle’s vertical leg, we can label our
figure as such:

Then starting with Pythagorean Theorem, we have

Then, since , we have

Notice that in order to have A(h)> 0 we need 0 <h< 21.6.

Graphing this function in our calculator, we find the maximum area of 44.9 cm^2,
when the

horizontal leg is 12.47 cm.

What if we had chosen to write our model for area in terms of the vertical leg
instead?

Then we would write ,and

Of course we still find the same maximum area of 44.9 cm^2, but with a vertical
leg length of

7.2 cm. Notice that while A(h) is simply a cubic function, A(v) is not even a
polynomial;

perhaps something different than what Precalculus students may be used to, and
thus explaining

the slightly different shapes of our earlier scatterplots of A(h) and A(v).
(Notice for A(v), the

“domain of reality is 0 <v< 21.6/2.

An Extension

What if we used a different-sized sheet of paper?

Redo the activity with various sizes of paper and collect data on the dimensions
that maximize

the triangle’s area. Some sample data follows:

Can we predict the horizontal leg length that maximizes the area from one of the
side lengths of

the paper?

**Horiz Leg_max (cm) vs. Shorter Dimension (cm)
**

Seems linear; let’s try a linear fit and check the residuals:

**Linear Fit**

Horiz Leg_max (cm) = 0.0257143 + 0.575928 Shorter Dimension (cm)

Looks good; what if we had tried to predict the horizontal leg length that maximizes the area

from the longer dimension?

**Horiz Leg_max (cm)**

While there seems to be some relationship; it’s not as clearly defined as the relationship between

the Horiz. Leg length and the shorter dimension.

So if h (x)=0.0257143 + 0.575928s (where h is the horizontal leg length that maximizes the

area and s is the length of the shorter dimension of the paper) is a good model to predict the

horizontal leg length, could we also model the vertical leg length from the shorter-side

dimension?

**Vert Leg_max (cm) vs. Shorter Dimension (cm)**

Looks linear based on the scatterplot; residuals confirm a good fit:

**Linear Fit**

Vert Leg_max (cm) = -0.041429 + 0.336333 Shorter Dimension (cm)

So, we seem to be able to predict the vertical leg length by v(s) = -0.041429 + 0.336333s,

where v is the vertical leg length that maximizes area and s is the length of the shorter dimension

of the paper.

**Possible extension:**

While the linear models for h(s) and v(s) seem very reasonable; it would perhaps be

advantageous for h(s) and v(s) to be “forced” to pass through (0,0), since a shorter-side

dimension of 0 cm would necessarily imply triangle legs of length 0. Then our models become

h(s)= 0.5776282s and v(s)= .3335938 with the following residual plots: (See Appendix 2

for a formula for this regression method.)

**Horiz Leg_max (cm) vs. Shorter Dimension (cm)**

**Linear Fit**

Horiz Leg_max (cm) = 0 + 0.5776282 Shorter Dimension (cm)

**Vert Leg_max (cm) vs. Shorter Dimension (cm)**

**Linear Fit**

Vert Leg_max (cm) = 0 + 0.3335938 Shorter Dimension (cm)

Finally, what if we try to predict the maximum area from the shorter paper dimension?

**Max Area (cm^2) vs. Shorter Dimension (cm)**

Could a linear model be a good fit? Not when we look at superimposed least-squares line and the

residual plot:

**Max Area (cm^2) vs. Shorter Dimension (cm)**

**Linear Fit**

Max Area (cm^2) = -17.29036 + 2.6948819 Shorter Dimension (cm)

Besides, if each leg is well-modeled by a linear function of the shorter dimension, then it seems

reasonable that the area should be modeled by a quadratic function of the shorter dimension:

**Max Area (cm^2) vs. Shorter Dimension (cm)**

**Polynomial Fit Degree=2**

Max Area (cm^2) = -18.81004 + 2.6948819 Shorter Dimension (cm) + 0.0942202 (Shorter Dimension (cm)-13.97)^2

Alternately, we could use re-expression to find a model for the area as a function of shorter

dimension. Since we think quadratic is a good model, we can take the square root of the area:

**sqrt(max area) vs. Shorter Dimension (cm)**

**Linear Fit**

sqrt(max area) = -0.012407 + 0.3112161 Shorter Dimension (cm)

When we write our model back in terms of the maximum area, we have:

(where a(s) is the maximum triangle area for a sheet of paper with shorter dimension s).

Again, as a possible extension, we might “force” our linear model through the origin (since it

makes sense that a shorter dimension of 0 cm would imply an area of 0). Then our model would

be:

Calculus method

Earlier we had

as a model to predict the area, A, as a function of horizontal leg length, h.
Noticing that the

“21.6” is simply the shorter dimension of the sheet of paper (in cm), we can
make our solution

more general by writing

where s is the shorter dimension of the sheet of paper. (Notice that, as we
observed earlier, the

maximum area is dependent only on the shorter dimension of the sheet of paper,
not the longer

dimension.)

Then we can find the value of h that maximizes A as follows:

Setting we have
, so .
(An easy check with the 2nd derivative

test verifies it is indeed a maximum.)

Alternately, if we had chosen to work with

instead, the algebra is slightly more challenging:

Then setting we have

Since and ,
it therefore follows that

Appendix 1

Excerpts from a sample student handout:

Triangle Problem Writing Assignment

o Write alone, but you may consult with others in your group.

**o Characteristics of your paper:**

o It should be “self-contained”, meaning that you could mail it to a friend back
home

who knows the math you know, and s/he would be able to follow your thinking.

o You can do it all by hand, or type the words and do the rest by hand, or learn

MathType and do the equations in a Word doc also.

o Include your name and the names of your pod-partners, a signature (which
indicates

that you were academically honest in doing this work), the name of the
assignment,

the date handed in, and your class block’s letter.

**o The paper should contain:**

o A description of the problem. (If you decide to take the description of the
problem

word-for-word from the handout, you must cite the source.)

o The “data solution”, which should include (with words wrapped around them, of

course):

The actual data, including units

A scatter plot (with axes labeled with words
and a scale)

An answer, as an ordered pair and in words

o The “algebra solution”, which should include:

Identifying your variables

The work leading up to your model

The model

A sketch of the model over the domain of
reality (and why that is the D.of.R)

An answer, as an ordered pair and in words

o A comparison of the two answers

If you want to say more (like, what you got by punching “magic buttons”, or
looking at the

relationship between base and height of the triangles), feel free

Appendix 2

Linear Regression with a “forced” intercept at 0.

The goal of regression is to minimize the sum of the squares of the residuals.
We start with

ordered pairs and the linear model y=mx.

Since a residual is “actual y” – “predicted y,” each residual can be written as
. So we set

out to minimize the sum of the squares of the residuals, or

To find the value of a that minimizes SSR(m), we take the derivative with
respect to m:

Setting the derivative equal to zero, we have

therefore

gives the slope of the line through the origin that minimizes the sum of the
squares of the

residuals.