A stick 5 cm long, a stick 9 cm long, and a third stick n cm long form a triangle. What is the sum of all possible whole number values of n?
9<n<14
Values of n: 10, 11, 12, 13.
Sum = 10+11+12+13 = 46 cm.
5<n<9
Values of n: 6, 7, 8.
Sum = 6+7+8 = 21 cm.
n = 5 cm(Isosceles Triangle).
n = 9 cm(Isosceles Triangle).
Sum Total=5+6+7+8+9+10+11+12+13 = 81 cm.
another of these triangle problems?
Have any ideas of your own on this one?
as a hint, suppose you joined the two sticks at the end.
Now consider rotating the smaller one, from being end-to-end with the larger one, to overlapping it. What's the max and min possible length for the third side?
Now list all the integers in that range and add 'em up.
I know the range is 14>x>4
and the third side can not be 5 or 9
so that leaves 6, 7, 8, 10, 11, 12,and 13. That added up all together is 67 but my teacher said that was incorrect
14+13+12+11+10+9+8+7+6+5+4=99
13+12+11+10+9+8+7+6+5=81
14+13+12+11+10+8+7+6=81
Which one is it?
Of course the 3rd side can be 5 or 9. That just makes an isosceles triangle. So, you want all the numbers n such that 4 < n < 14:
5+6+7+8+9+10+11+12+13 = 81
I say use the Pythagorean theorem.
a^2+b^2=C^2.
5^2+9^2=C^2
25+81=106
THEN DO THE SQUARE ROOT OF 106 WHICH IS 10.30CM
To determine the sum of all possible whole number values of n, we need to find the range of values for n that can form a valid triangle with the given sticks.
In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Let's check this condition for each possible combination of the sticks:
1. Stick 5 cm + Stick 9 cm > Stick n cm:
5 + 9 > n
14 > n
2. Stick 5 cm + Stick n cm > Stick 9 cm:
5 + n > 9
n > 4
3. Stick 9 cm + Stick n cm > Stick 5 cm:
9 + n > 5
n > -4
We have found two conditions from the inequalities above:
1. n > 4
2. n > -4
Combining these two conditions, we conclude that n must be greater than 4 to form a valid triangle.
Now, since the question asks for the sum of all possible whole number values of n, we need to find the range of whole number values greater than 4:
The whole number values greater than 4 are 5, 6, 7, 8, 9, 10, ...
To find the sum of this arithmetic sequence, we can use the formula for the sum of an arithmetic series:
Sum = (n/2) * (first term + last term)
Here, the first term is 5 and the last term is infinity (∞), as there is no upper limit given. So, the sum is:
Sum = (n/2) * (5 + ∞)
Since infinity (∞) is not a finite number, we cannot obtain a definite sum. Therefore, we cannot calculate the sum of all possible whole number values of n in this case.