Matilde wants to build a wooden planter in the shape of a right triangle for their garden. One side will

Question

be formed by an existing bench that is 96 inches long. To fence the other two sides, Matilde wants to
use pieces of wood that are each precut to 12 inches, so each of the two undetermined side lengths
will need to be in whole number of inches. Use the polynomial identity (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2ches. Use the polynomial identity
to find one of the other possible side lengths, assuming 2xy=96
and that the two undetermined sides are of different lengths. Which of the following values
could be one of the side lengths for their wooden planter?
24 inches
100 inches
20 inches
80 inches

Answer 1

To solve the problem, we are given that \( 2xy = 96 \). From this, we can determine:

\[ xy = \frac{96}{2} = 48 \]

Now, we can utilize the values we obtained in conjunction with the polynomial identity \( (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 \). We already know that \( 2xy = 96 \), so we can calculate \( (2xy)^2 \):

\[ (2xy)^2 = 96^2 = 9216 \]

Next, we need to express \( x^2 + y^2 \) in terms of \( x \) and \( y \). We also know \( xy = 48 \). Therefore, we can find \( x^2 + y^2 \) using the identity:

\[ (x + y)^2 = x^2 + y^2 + 2xy \]

Thus, rearranging gives:

\[ x^2 + y^2 = (x + y)^2 - 2xy \]

We will also need \( x^2 - y^2 \):

\[ (x - y)^2 = x^2 + y^2 - 2xy \]

Now we can denote \( p = x + y \) and \( q = x - y \):

1. \( p^2 = x^2 + y^2 + 2xy = x^2 + y^2 + 96 \)
2. \( q^2 = x^2 + y^2 - 2xy = x^2 + y^2 - 96 \)

Plugging these back into the polynomial identity:

\[ (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 \]

We can substitute \( 2xy = 96 \):

\[ (x^2 + y^2)^2 = q^2 + 9216 \]

Now, let’s plug in \( xy = 48 \) to find possible values for \( x \) and \( y \). The possible pairs \( (x, y) \) must satisfy:

1. \( x + y = p \)
2. \( xy = 48 \)

We can express \( y \) in terms of \( x \):

\[ y = \frac{48}{x} \]

Then we want \( 2xy = 96 \), which gives:

\[ x + y = p = x + \frac{48}{x} \]

This means we need to consider \( x \) such that both \( x + \frac{48}{x} \) yields integer results.

Testing integer factors of 48 for \( x \):

• \( x = 1 \), \( y = 48 \) → \( 1 + 48 = 49 \)
• \( x = 2 \), \( y = 24 \) → \( 2 + 24 = 26 \)
• \( x = 3 \), \( y = 16 \) → \( 3 + 16 = 19 \)
• \( x = 4 \), \( y = 12 \) → \( 4 + 12 = 16 \)
• \( x = 6 \), \( y = 8 \) → \( 6 + 8 = 14 \)

The possible side lengths come from knowing \( x \) and \( y \). The lengths would consist of combinations that also fulfill the condition that \( (x^2 + y^2) \) remains reasonable against integer outputs.

By observing \( (96 - xy) \) or others from this polynomial connection, 24 inches or 48 inches related directly.

Out of the answer choices provided:

• 24 inches is a different length than initially preset.

Verifying whether 24 is a usable length for \( x \) or \( y \):

Given all that, the only answer from the choices Matilde could consider using operative integers would be:

24 inches.