A circular flower bed is sectioned off. The arc intercepted by the 120 degree central angle is to be bordered with marigolds. The garden has a diameter of 63 inches. Circumference = Pi * diameter, Pi = 22/7. If you plant the marigolds about eight inches apart what is the least number you must buy?
The radius of the circle is given by
2*pi/3 * r = 63
r = 30
So, the area of the 1/3 of the flower bed to be planted is
pi/3 * r^2 = pi/3 * 900 = 943 sq in.
If the marigolds are planted in a rectangular grid, that allocates 64 sq in per flower, so you'd need about
943/64 = 15 marigolds.
Of course, an 8" square grid means that diagonally opposite flowers are about 11.2" apart. If you plant them in a triangular lattice of side 8", then you'd have somewhere 20 around flowers, depending on how they were arranged
To find the least number of marigolds you must buy, you need to determine the length of the arc intercepted by the 120-degree central angle.
First, calculate the circumference of the circular flower bed using the given diameter of 63 inches:
Circumference = π * Diameter
Circumference = (22/7) * 63
Circumference ≈ 198 inches
To find the length of the arc intercepted by a 120-degree central angle, you can use the formula:
Arc Length = (Central Angle / 360) * Circumference
Plugging in the values:
Arc Length = (120 / 360) * 198
Arc Length = (1/3) * 198
Arc Length ≈ 66 inches
Since you intend to plant the marigolds about eight inches apart, divide the length of the arc by the spacing between the marigolds to determine the number of marigolds you need to buy:
Number of Marigolds = Arc Length / Spacing
Number of Marigolds = 66 / 8
Number of Marigolds ≈ 8.25
Since you cannot buy a fractional amount of marigolds, you will need to round up to the nearest whole number. Therefore, you must buy at least 9 marigolds to border the arc intercepted by the 120-degree central angle.