Question

Part a. The radius of circle N is 3 inches, and the radius of circle O is 4 inches. What is the radius of circle P whose area is the sum of the areas of circles N and O?

part b. A small square has side length 7 cm, and a medium square has side length 24 cm. What is the side length of a large square whose area is the sum of the areas of the small and medium squares?



part c. The side length of a small equilateral triangle is 10 feet, and the side length of a medium equilateral triangle is 24 feet. What is the side length of a large equilateral triangle whose area is the sum of the areas of the small and medium equilateral triangles?

Answer 1

Explanation:

part a. The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.

The area of circle N is:

A_N = π(3 inches)^2 = 9π square inches

The area of circle O is:

A_O = π(4 inches)^2 = 16π square inches

The area of circle P, whose area is the sum of the areas of circles N and O, is:

A_P = A_N + A_O = 9π + 16π = 25π square inches

To find the radius of circle P, we can use the formula for the area of a circle and solve for the radius:

A_P = πr^2

25π = πr^2

r^2 = 25

r = 5 inches

Therefore, the radius of circle P is 5 inches.

part b. The area of a square is given by the formula A = s^2, where A is the area and s is the side length.

The area of the small square is:

A_small = (7 cm)^2 = 49 square cm

The area of the medium square is:

A_medium = (24 cm)^2 = 576 square cm

The area of the large square, whose area is the sum of the areas of the small and medium squares, is:

A_large = A_small + A_medium = 49 + 576 = 625 square cm

To find the side length of the large square, we can use the formula for the area of a square and solve for the side length:

A_large = s^2

625 = s^2

s = 25 cm

Therefore, the side length of the large square is 25 cm.

part c. The area of an equilateral triangle is given by the formula A = (sqrt(3)/4)*s^2, where A is the area and s is the side length.

The area of the small equilateral triangle is:

A_small = (sqrt(3)/4)*(10 feet)^2 = 25sqrt(3) square feet

The area of the medium equilateral triangle is:

A_medium = (sqrt(3)/4)*(24 feet)^2 = 144sqrt(3) square feet

The area of the large equilateral triangle, whose area is the sum of the areas of the small and medium equilateral triangles, is:

A_large = A_small + A_medium = 25sqrt(3) + 144sqrt(3) = 169sqrt(3) square feet

To find the side length of the large equilateral triangle, we can use the formula for the area of an equilateral triangle and solve for the side length:

A_large = (sqrt(3)/4)*s^2

169sqrt(3) = (sqrt(3)/4)*s^2

s^2 = 676

s = 26 feet (since s is a positive length)

Therefore, the side length of the large equilateral triangle is 26 feet.