Matilde wants to build a wooden planter in the shape of a right triangle for their garden. One side will 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 (x2+y2)2=(x2−y2)2+(2xy)2 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?(1 point)
Responses
20 inches
20 inches
80 inches
80 inches
100 inches
100 inches
24 inches
24 inches
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The identity provided is used to calculate the possible side lengths of the right triangle. Given that 2xy = 96 and the two undetermined sides are different lengths, we can rewrite the identity as:
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2
Plugging in 2xy = 96:
(x^2 + y^2)^2 = (x^2 - y^2)^2 + 96^2
(x^2 + y^2)^2 = (x^2 - y^2)^2 + 9216
Let's test the options provided:
If we try x = 20 and y = 4:
(20^2 + 4^2)^2 = (20^2 - 4^2)^2 + 9216
(400 + 16)^2 = (400 - 16)^2 + 9216
(416)^2 = (384)^2 + 9216
(6)(231) = (13)(297) + 9216
1496 = 3861 + 9216
1496 = 13077
This test shows that 20 inches is not a possible side length.
Let's try x = 80 and y = 6:
(80^2 + 6^2)^2 = (80^2 - 6^2)^2 + 9216
(6400 + 36)^2 = (6400 - 36)^2 + 9216
(6436)^2 = (6364)^2 + 9216
(9)(103)(389) = (4)(158)(420) + 9216
35937 = 26520 + 9216
35937 = 35736
This shows that 80 inches is not a possible side length.
Let's try x = 100 and y = 8:
(100^2 + 8^2)^2 = (100^2 - 8^2)^2 + 9216
(10000 + 64)^2 = (10000 - 64)^2 + 9216
(10064)^2 = (99636)^2 + 9216
(4)(1609) = (4)(201)(1469) + 9216
6436 = 5874 + 9216
6436 = 6436
This shows that 100 inches could be one of the possible side lengths.
Therefore, the side length that could be one of the side lengths for their wooden planter is 100 inches.