An equilateral triangle is inscribed in a circle of radius 6r. Express the area A within the circle but outside the triangle as a function of the length 5x of the side of the triangle.

Step-by-step explanation:

We have been given that an equilateral triangle is inscribed in a circle of radius 6r. We are asked to express the area A within the circle but outside the triangle as a function of the length 5x of the side of the triangle.

We know that the relation between radius (R) of circumscribing circle to the side (a) of inscribed equilateral triangle is .

Upon substituting our given values, we will get:

Let us solve for r.

We know that area of an equilateral triangle is equal to , where s represents side length of triangle.

The area within circle and outside the triangle would be difference of area of circle and triangle as:

We can make a common denominator as:

Therefore, our required expression would be .

D. 135.37 in²

Step-by-step explanation:

We have that the gift paper consists of two circles and a square.

Thus, we will need to find the area of the circles and the square.

Now, as the diameter of the circle = 4 inches

So, the radius of the circle = = 2 inches

Since, the area of a circle =

This implies that area of the two circles =

i.e. Area of the circles =  

i.e. Area of the circles =  

i.e. Area of the two circles = 25.133 in²

Further, one side of the square = 10.5 inches

Thus, the area of the square =

i.e. Area of the square =

i.e. Area of the square = 110.25 in²

So, the overall area of the gift paper = 25.133 + 110.25 = 135.37 in².

Hence, the amount of paper used by Angela is 135.37 in².