two sides of an equilateral trianlge measure (2y+3) units and (y^2-5) units. if the perimeter of the triangle is 33 units, what is the value of y?
To solve this problem, we'll set up and solve an equation based on the given information.
An equilateral triangle has all three sides of equal length. So, if two sides of the triangle measure (2y+3) units and (y^2-5) units, then the third side must also be equal in length.
Let's represent the length of the third side as z.
The perimeter of a triangle is the sum of the lengths of its three sides. In this case, the perimeter is given as 33 units. Therefore, we can write the equation:
(2y + 3) + (y^2 - 5) + z = 33
Now, let's simplify the equation:
2y + 3 + y^2 - 5 + z = 33
Combine like terms:
y^2 + 2y + z - 2 = 33
Rearrange the equation:
y^2 + 2y + z = 35 [Equation 1]
Since we know that the triangle is equilateral, we also know that the three sides are equal in length. So, we can set up another equation:
(2y + 3) = (y^2 - 5)
Simplify further:
2y + 3 = y^2 - 5
Rearrange the equation:
y^2 - 2y - 8 = 0
To solve this quadratic equation, we can factor it:
(y - 4)(y + 2) = 0
Setting each factor equal to zero:
y - 4 = 0 ---> y = 4
y + 2 = 0 ---> y = -2
Since the length of a side cannot be negative, we can disregard the solution y = -2.
Therefore, the value of y is 4.