Sue is building a triangle out of sticks. The length of the first side is "a" and the second side is of length "b". The length of the third side must be between 8 and 26 units. Find the product of "a" and "b".
we know that 8 < a+b < 26, and if a>b,
c-b < a < c+b
Other than that, there's not much to go on.
For example, we could have
c a b
8 4 6
13 9 9
and lots of others.
Is there something not explained?
To find the product of "a" and "b", we need to consider the conditions for a triangle.
In a triangle, any two sides must add up to be greater than the length of the third side. Additionally, the difference between any two sides must be less than the length of the third side.
Let's consider the conditions given:
1. The length of the first side is "a".
2. The length of the second side is "b".
3. The length of the third side must be between 8 and 26 units.
For a triangle to be formed, the sum of the lengths of the two shortest sides must be greater than the length of the third side. So we have two conditions to satisfy:
1. a + b > third side
2. b + third side > a
Since the length of the third side must be between 8 and 26 units, we can write the following inequality:
8 ≤ third side ≤ 26
Now, let's consider the values that "a" and "b" can take. Since we want to find the product of "a" and "b", we don't have specific values for them. Instead, we need to find the range of values they can take.
From the conditions above, we know that both "a" and "b" should be greater than 0. So let's assume a > 0 and b > 0.
Now, we can combine the conditions to find the range of values for "a" and "b":
1. a + b > 8
2. b + 8 > a
3. a + b < 26
4. 26 - b > a
To simplify these conditions, we can consider the worst-case scenario: when the third side has its maximum value of 26.
From condition 1: a + b > 8 ⟹ b > 8 - a
From condition 2: b + 8 > a ⟹ b > a - 8
From condition 4: 26 - b > a ⟹ b < 26 - a
Combining the inequalities, we get:
8 - a < b < 26 - a
Now, we need to consider the case where a and b are integers, as we are dealing with the length of sticks.
To find the range of values, we can try different integer values for "a" and "b" within the given constraints and see which combinations satisfy the inequalities.
For example, if we assume a = 1, then the range of valid values for b would be:
8 - 1 < b < 26 - 1
7 < b < 25
Similarly, for a = 2, the range of valid values for b would be:
7 < b < 24
We can continue this process and find the range of values for "b" for each value of "a" within the given conditions. Once we have all the valid combinations, we can calculate the product of "a" and "b".
So, without knowing the specific values of "a" and "b", it is not possible to find their product. We need to analyze the range of values based on the given conditions.