|<---- 20 cm ---->|
| |
+------------------+
|<---- 41 cm ---->|
| |
+------------------+
|<---- 44 cm ---->|
| |
+------------------+
After x is cut off each rod, the remaining lengths can be arranged to form a right triangle, as shown below:
+---------(44-x)--------+
| |
| |
| |
| |
| |
| |
(20-x) | | (41-x)
+-----------+------------------------+
| x |
b) To model the situation, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. In this case, we have:
(20-x)^2 + (41-x)^2 = (44-x)^2
c) To find the length that is cut off, we can solve the equation from part (b) for x. First, we can simplify the equation by expanding the squares:
400 - 40x + x^2 + 1681 - 82x + x^2 = 1936 - 88x + x^2
Simplifying further, we get:
2x^2 - 18x - 855 = 0
We can solve for x by using the quadratic formula:
x = [18 ± sqrt(18^2 + 4(2)(855))] / (2(2))
x = [18 ± sqrt(7404)] / 4
x ≈ 16.98 or x ≈ -24.98
Since x represents a length that is cut off each rod, it must be positive. Therefore, we can discard the negative solution and conclude that the length that is cut off is approximately 16.98 cm.
d) Using the length that is cut off, we can find the dimensions of the right triangle by substituting x = 16.98 into the expressions for the remaining lengths. We get:
(20 - 16.98) = 3.02 cm
(41 - 16.98) = 24.02 cm
(44 - 16.98) = 27.02 cm
Therefore, the dimensions of the right triangle are 3.02 cm, 24.02 cm, and 27.02 cm.