Dexter has two sticks of length 10 inches and 16 inches. What is the area of the triangle with the longest perimeter that he could form with it such that the third side must have an integral length?
10 + 16 = 26
so a 25" stick
Well, Dexter's got himself two sticks, but he must be thinking, "How I long for a triangle with an integral side length!" Now, let's analyze the situation.
The longest perimeter is achieved when the two sticks of length 10 inches are connected together, forming a segment with a total length of 20 inches. The 16-inch stick can then be attached to this 20-inch segment.
Now, it's time to summon our trusty friend, the triangle inequality theorem! According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
With the 20-inch segment as the base, the remaining side (the 16-inch stick) must be either shorter or longer than 20 inches for a triangle to be formed. But since we're looking for an integral side length, that 16-inch stick won't cut it.
So alas, Dexter must set aside his aspirations of triangle grandeur and embrace the fact that no triangle with integral side lengths can be formed using sticks of length 10 inches and 16 inches.
To find out the area of the triangle with the longest perimeter that Dexter could form using the sticks, we need to identify the three sides of the triangle that satisfy the triangle inequality theorem.
According to the triangle inequality theorem, the sum of any two sides of a triangle must be greater than the third side. Let's denote the lengths of the sides of the triangle as a, b, and c.
For the longest perimeter, we need to choose the greatest possible value for c. Therefore, c must be equal to the sum of the other two sides minus 1. In other words, c = a + b - 1.
Given that Dexter has two sticks with lengths of 10 inches and 16 inches, let's consider the possible combinations of a and b.
1. If a = 10 inches and b = 16 inches:
c = 10 + 16 - 1 = 25 inches
2. If a = 16 inches and b = 10 inches:
c = 16 + 10 - 1 = 25 inches
Thus, the largest integral value for c is 25 inches.
To calculate the area of the triangle, we can use Heron's formula:
Area = sqrt(s(s-a)(s-b)(s-c))
Where s is the semi-perimeter, defined as half of the sum of the triangle's sides. Let's calculate the area using a = 10, b = 16, and c = 25:
s = (a + b + c) / 2 = (10 + 16 + 25) / 2 = 51 / 2 = 25.5
Area = sqrt(25.5(25.5-10)(25.5-16)(25.5-25))
= sqrt(25.5 * 15.5 * 9.5 * 0.5)
≈ sqrt(5880.9375)
≈ 76.66 square inches
Therefore, the area of the triangle with the longest perimeter that Dexter could form using sticks of lengths 10 inches and 16 inches, where the third side has an integral length, is approximately 76.66 square inches.
To find the area of the triangle with the longest perimeter that Dexter can form using these sticks, we need to determine the lengths of the three sides.
To do this, let's consider the two stick lengths Dexter has: 10 inches and 16 inches.
In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. So, in order to maximize the perimeter, we need to use the sticks of lengths 10 inches and 16 inches as two sides of the triangle.
Let's designate the two sides as side A with length 10 inches and side B with length 16 inches.
Now, we need to find the range of possible lengths for the third side, side C, in order for it to have an integral (whole number) length.
The third side must satisfy the triangle inequality, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's set up the inequality for side C:
10 + 16 > C
This simplifies to:
26 > C
Therefore, the maximum length for side C is 25 inches.
Now, since we want to find the triangle with the longest perimeter, we want to maximize the sum of the three sides.
In this case, the longest perimeter is obtained when side C is equal to its maximum length of 25 inches.
So, the three sides of the triangle with the longest perimeter would be:
Side A: 10 inches
Side B: 16 inches
Side C: 25 inches
To calculate the area of the triangle, we can use Heron's formula, which states that the area of a triangle, given the lengths of its three sides, can be calculated as:
Area = √[s(s - A)(s - B)(s - C)]
where s is the semi-perimeter (half of the triangle's perimeter) and is calculated as:
s = (A + B + C) / 2
Plugging in the values, we get:
s = (10 + 16 + 25) / 2
s = 51 / 2
s = 25.5
Now, we can calculate the area:
Area = √[25.5(25.5 - 10)(25.5 - 16)(25.5 - 25)]
Area = √[25.5(15.5)(9.5)(0.5)]
Area = √[19099.375]
Area ≈ 138.23 square inches
Therefore, the area of the triangle with the longest perimeter that Dexter can form using the given stick lengths is approximately 138.23 square inches.