Final answer:
The central angle of the sector with an area of 16π/5 ft² and a 12-foot diameter circle is 128 degrees.
Step-by-step explanation:
To find the measure of the central angle of the sector whose area is 16π/5 ft² and the diameter of the circle is 12 feet, we will use the formula for the area of a sector:
A = (θ/360) * π * r²
where A is the area of the sector, θ is the central angle in degrees, and r is the radius of the circle. Given that the diameter is 12 feet, the radius (r) is half of the diameter, so r = 6 feet. We can solve for θ:
16π/5 = (θ/360) * π * (6²)
16π/5 = (θ/360) * π * 36
θ = (16π/5) * (360/(π * 36))
θ = (16 * 360)/(5 * 36)
θ = 128 degrees
Therefore, the measure of the central angle of the sector is 128 degrees, which corresponds to answer choice D).