Question

[Mass and Density] Consider a thin metal disk in R² which occupies the part of the disk x²+y²≤9 in the first quadrant. The disk has a variable density (in g/m²) of rho(x,y)=1+2x+3y. (a) What point(s) of the disk have the highest density? the lowest density? (b) What is the total mass of the disk? (c) What is the average density of the disk? (d) Is the average density just equal to the average of the highest and lowest density?

Answer 1

To find the highest and lowest density points on the metal disk, evaluate the density function at different points. For the total mass, integrate the density function over the disk. The average density is not equal to the average of the highest and lowest density.

Step-by-step explanation:

To find the highest and lowest density points on the metal disk, we need to evaluate the density function at different points. The density function is given as rho(x, y) = 1 + 2x + 3y. To find the highest density point, we need to maximize this function, and to find the lowest density point, we need to minimize this function.

To find the total mass of the disk, we integrate the density function over the disk. The disk is defined by the inequality x² + y² ≤ 9 in the first quadrant, so we need to integrate the density function over this region.

The average density of the disk can be found by dividing the total mass of the disk by the area of the disk.

No, the average density is not equal to the average of the highest and lowest density. The average density considers the distribution of mass over the entire disk, while the highest and lowest density points only consider the local density at those specific points.