Final answer:
To find m∠DCE, we calculate the fraction of the total area of the circle that the shaded sector represents and apply this fraction to the full 360 degrees of the circle, resulting in an angle measure of 80 degrees.
Step-by-step explanation:
The student is asked to find the measure of angle DCE in a circle with center C, where CD is a radius of length 2 and the area of the shaded sector is 8/9 π.
First, we need to compute the total area of the circle using the formula A = πr^2, where r is the radius. Since CD equals 2, the radius r is also 2, which gives us an area of 4π for the entire circle.
Next, we find the fraction of the circle that the shaded sector represents by dividing the area of the sector by the total area of the circle, which is 8/9π / 4π = 2/9.
Since the area of a sector is a fraction of the whole circle's area, this same fraction applies to the central angle. Therefore, the measure of m∠DCE is 2/9 of the full circle's 360 degrees, which is 2/9 × 360 = 80 degrees.