We have a circle with an inscribed square with an inscribed circle with an inscribed equilateral triangle. The diameter of the outer circle is 8 feet. The triangle material cost $104.95 a square foot. What is the cost of the triangular center?

Answer 1

The cost of a triangular center is $1090.67

#AC = 8# as a given diameter of a circle.
Therefore, from the Pythagorean Theorem for the right isosceles triangle #Delta ABC#, #AB = 8/sqrt(2)#
Then, since #GE = 1/2 AB#, #GE = 4/sqrt(2)#
Obviously, triangle #Delta GHI# is equilateral.
Point #E# is a center of a circle that circumscribes #Delta GHI# and, as such is a center of intersection of medians, altitudes and angle bisectors of this triangle. It is known that a point of intersection of medians divides these medians in the ratio 2:1 (for proof see Unizor and follow the links Geometry - Parallel Lines - Mini Theorems 2 - Teorem 8)
Therefore, #GE# is #2/3# of the entire median (and altitude, and angle bisector) of triangle #Delta GHI#.
So, we know the altitude #h# of #Delta GHI#, it is equal to #3/2# multiplied by the length of #GE#: #h = 3/2 * 4/sqrt(2) = 6/sqrt(2)#
Knowing #h#, we can calculate the length of the side #a# of #Delta GHI# using the Pythagorean Theorem: #(a/2)^2+h^2=a^2# from which follows: #4h^2=3a^2# #a=(2h)/sqrt(3)#
Now we can calculate #a#: #a = (2*6)/(sqrt(2)*sqrt(3)) =2sqrt(6)#
The area of a triangle is, therefore, #S = 1/2ah = 1/2*2sqrt(6)*6/sqrt(2) = 6sqrt(3)#
At a price of $104.95 per square foot, the price of a triangle is #P = 104.95*6sqrt(3)~~1090.67#
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Answer 2

To find the cost of the triangular center, we first need to determine the side length of the equilateral triangle inscribed within the circle.

The diameter of the outer circle is 8 feet, so the radius of the outer circle is ( \frac{8}{2} = 4 ) feet.

The side length of the equilateral triangle inscribed in the circle is equal to twice the radius of the outer circle, which is ( 2 \times 4 = 8 ) feet.

To find the area of an equilateral triangle, we use the formula ( A = \frac{\sqrt{3}}{4} \times \text{side length}^2 ).

Substituting the side length ( 8 ) feet into the formula, we get:

[ A = \frac{\sqrt{3}}{4} \times 8^2 = \frac{\sqrt{3}}{4} \times 64 = 16\sqrt{3} , \text{square feet} ]

The cost of the triangular center is the area of the triangle multiplied by the cost per square foot:

[ \text{Cost} = 16\sqrt{3} \times $104.95 = $1679.20 ]

So, the cost of the triangular center is $1679.20.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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