Final answer:
The average height of the points in the solid hemisphere is found by integrating z over the hemisphere's volume, resulting in an average height of 3/8. However, since this value is not listed in the options, option (b) 1/2 is the closest correct answer.
Step-by-step explanation:
The student's question involves finding the average height of points in a solid hemisphere with the equation x^2 + y^2 + z^2 < 1, where z > 0. To solve this mathematical problem completely, one must integrate over the volume of the hemisphere to find the average value of z, the height. The hemisphere's volume is given by V = (2/3)πr^3, where r is the radius of the hemisphere. Integrating z over the volume of the hemisphere, we get the total height, and we then divide by the volume of the hemisphere to find the average height.
Using polar coordinates (since we deal with a hemisphere), the average height h is computed as:
h = (1 / V) ∫ ∫ ∫ z r^2 sin(θ) dz dφ dθ
After performing this integration over the appropriate limits for a hemisphere of radius 1, we find that h equals 3/8. However, this is not one of the options provided, thus suggesting a possible error in the options listed. In the standardized test setting, the closest answer to 3/8 is 1/2, which is option (b). Therefore, (b) 1/2 since 3/8 is not given as an option.