an equilateral triangle has an inscribed circle with a radius of sqrt b+27 ft. if the area of the triangle is 48*sqrt b+6 feet squared. what are the two possible value of perimeter for this triangle?

To find the two possible values of the perimeter for the equilateral triangle, we need to use the given information about the inscribed circle and the area of the triangle.

Let's start by finding the side length of the equilateral triangle. The formula to find the area of an equilateral triangle is:

Area = (sqrt(3)/4) * side^2

Given that the area of the triangle is 48 * sqrt(b+6) ft^2, we can set up the equation:

48 * sqrt(b+6) = (sqrt(3)/4) * side^2

Next, we need to find the side length of the triangle. We can solve this equation for 'side':

side^2 = (48 * sqrt(b+6)) / (sqrt(3)/4)

Simplifying the right side of the equation:

side^2 = (48 * sqrt(b+6)) * (4/sqrt(3))

side^2 = 192 * sqrt(b+6) / sqrt(3)

Now, let's determine the side length of the equilateral triangle:

side = sqrt(192 * sqrt(b+6) / sqrt(3))

Since it is an equilateral triangle, all three sides will have the same length.

The perimeter of the triangle is given by:

Perimeter = 3 * side

Substituting the value of 'side' we found earlier:

Perimeter = 3 * sqrt(192 * sqrt(b+6) / sqrt(3))

Now, we need to consider the inscribed circle's radius, which is sqrt(b+27) ft. This radius is equal to the distance from the triangle's center to any of its vertices.

In an equilateral triangle, the distance from the center to any vertex is also equal to one-third of the length of the median. The median bisects the side of the triangle and is also the radius of the inscribed circle. Therefore, we have:

side = 3 * sqrt(b+27)

Now, we have two equations for the side length of the triangle:

1) side = sqrt(192 * sqrt(b+6) / sqrt(3))
2) side = 3 * sqrt(b+27)

We can set these two equations equal to each other to solve for 'b':

sqrt(192 * sqrt(b+6) / sqrt(3)) = 3 * sqrt(b+27)

Squaring both sides of the equation:

(192 * sqrt(b+6) / sqrt(3)) = 9 * (b+27)

Simplifying further:

192 * sqrt(b+6) = 9 * sqrt(3) * (b+27)

Squaring both sides of the equation again:

(192 * sqrt(b+6))^2 = (9 * sqrt(3) * (b+27))^2

Simplifying and solving for 'b':

(192 * (b+6)) = (9 * sqrt(3) * (b+27))

Then, we can solve for 'b':

b = (11664 * sqrt(3)) / (9 * sqrt(3) - 192)

Now, substitute this value of 'b' in both equations for the side length of the triangle. Calculate each value, and you will have the two possible perimeters of the equilateral triangle.