Benny wants to plant a garden around the base of a tree. To determine the amount of topsoi needed, he measured the circumference of the tree and found it to be 9.5 ft. His garden will be 2.0 feet wide in the form of a ring around the tree. Find to the nearest square ft. the surface area of the garden Benny intends to plant
the circumference of the inner circle is 9.5 feet which is 2 pi Rtree where Rtree is the radius of the tree.
From that find
Rtree = 9.5 /(2*pi)
then you can find the area of the tree cross section
Atree = pi Rtree^2
Then
the outer radius of the path is
Rout = Rtree + 2
and the area of the big circle, which includes the tree cross section is:
pi Rout^2
the difference is the area of the garden
pi(Rout^2 - Rtree^2)
To find the surface area of the garden Benny intends to plant, we need to calculate the area of the ring around the tree.
First, let's find the radius of the ring. The circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius. Rearranging the formula to solve for r, we have r = C / (2π).
In this case, the circumference of the tree is given as 9.5 ft. So, the radius of the tree is:
r = 9.5 ft / (2π)
r ≈ 9.5 ft / (2 * 3.14)
r ≈ 1.51 ft
Since the garden will be 2.0 feet wide, the outer radius of the ring will be r + 2, and the inner radius of the ring will be r.
Outer radius = 1.51 ft + 2 ft = 3.51 ft
Inner radius = 1.51 ft
Now, we can calculate the area of the ring using the formula for the area of a ring. The formula is A = π(R^2 - r^2), where A is the area, R is the outer radius, and r is the inner radius.
Area of the ring = π((3.51 ft)^2 - (1.51 ft)^2)
Area of the ring ≈ 3.14 * (12.32 ft^2 - 2.28 ft^2)
Area of the ring ≈ 3.14 * 10.04 ft^2
Area of the ring ≈ 31.5344 ft^2
Therefore, the surface area of the garden Benny intends to plant is approximately 31.5344 square feet. Rounded to the nearest square foot, the surface area is 32 square feet.