Question

The blueprint below represents a circular garden. Nicole wants to put a pond in sector ACB as marked on the blueprint. The radius of the garden is 3 meters. What area of the garden will be covered by the pond? Round to the nearest tenth.

Answer 1

Explanation:

To find the area of the garden covered by the pond, we need to find the area of sector ACB.

First, we need to find the measure of the central angle of sector ACB. The central angle is the same as the angle formed by radii OA and OB.

Since the radius of the garden is 3 meters, we can use the Pythagorean theorem to find the length of the chord AB:

AB² = OA² + OB²

AB² = 3² + 3²

AB² = 18

AB = √18 ≈ 4.24 meters

Now, we can use the formula for the area of a sector:

A = (θ/360)πr²

where θ is the central angle and r is the radius of the circle.

The central angle of sector ACB can be found using the inverse cosine function:

cos(θ/2) = AB/2r

cos(θ/2) = 4.24/(2*3)

cos(θ/2) ≈ 0.707

θ/2 ≈ cos⁻¹(0.707)

θ ≈ 2cos⁻¹(0.707)

θ ≈ 144.1 degrees

Now we can calculate the area of sector ACB:

A = (θ/360)πr²

A = (144.1/360)π(3)²

A ≈ 10.6 square meters

Therefore, the area of the garden covered by the pond is approximately 10.6 square meters.