Two sides of a triangle measure 14 and 10. If A is the set of possible lengths for the third side, how many elements of A are integers?
Is the correct answer 19 elements of A are integers?
To determine the number of elements in set A that are integers, we need to find the possible lengths for the third side of the triangle given that two sides measure 14 and 10.
According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
In this case, the two given sides measure 14 and 10. Let's denote the length of the third side as x.
Using the triangle inequality theorem, we have two inequalities:
10 + x > 14 (x + 10 is greater than 14)
14 + x > 10 (x + 14 is greater than 10)
Simplifying these inequalities, we get:
x > 4 (subtracting 10 from both sides of the first inequality)
x > -4 (subtracting 14 from both sides of the second inequality)
Since we are looking for integer solutions, we can conclude that the possible lengths for the third side lie in the set A = {x | x > 4 and x > -4}.
To find the number of elements in set A, we need to find the interval where x can exist.
The inequality x > 4 represents all values of x greater than 4, including decimals and fractions.
The inequality x > -4 represents all values of x greater than -4.
Since the inequalities are not limited to integer values, there will be infinitely many elements in set A.
Therefore, the correct answer is that there are infinitely many elements in set A that are integers, not 19 elements.