Solve the problem.
-An industrial plant located in the center of a small town emits particulate matter into the atmosphere. Suppose the concentration of particulate matter in parts per million at a point d miles from the plant is given by
C = 120 - 15d2. If the boundaries of the town form a rectangle four miles long and six miles wide, what is the
Average concentration of particulate matter throughout the city? Express C as a function of x and y, set up a
Double integral, and evaluate.

Question

Quizplus · Accepted Answer

We need to find the average concentration of particulate matter throughout the town, which requires finding the double integral of the concentration function over the town area and dividing it by the area.
To express C as a function of x and y, we need to rewrite d in terms of x and y. Since the town is a rectangle and the plant is located at the center, we can use the distance formula to find d:
d = sqrt((x-2)^2 + (y-3)^2)
Substituting this into the expression for C, we get:
C(x,y) = 120 - 15((x-2)^2 + (y-3)^2)
To find the double integral, we integrate C over the rectangle [0,4] x [0,6]:
∬C(x,y) dA = ∫0^4 ∫0^6 (120 - 15((x-2)^2 + (y-3)^2)) dy dx
This is a straightforward integration using the standard integral forms. Evaluating this double integral, we get:
∬C(x,y) dA = 1680
Finally, we divide by the area of the town to find the average concentration:
Average concentration = 1680 / (4*6) = 70
Therefore, the answer is B. We want to find the average concentration of particulate matter throughout the entire town. 
The concentration at a point (x,y) can be given by the function C = 120 - 15d^2, where d is the distance between the point (x,y) and the industrial plant. Using the distance formula, we have

d = sqrt((x-2)^2 + (y-3)^2)

Note that the industrial plant is located at the point (2,3) since it is in the center of the town.

To find the average concentration throughout the town, we need to find the double integral of C over the entire town and divide by the area of the town, which is 4 x 6 = 24 square miles.

So we have

Average concentration = (1/24) * double integral over the entire town of (120 - 15((x-2)^2 + (y-3)^2)) dA

We can evaluate this integral by converting it to polar coordinates. The bounds of integration for r will be 0 to 5 (the distance from the lower left corner of the rectangle to the upper right corner), and the bounds of integration for theta will be 0 to pi/2 (since we only need to integrate over the first quadrant of the rectangle).

So the integral becomes

(1/24) * integral from 0 to pi/2 integral from 0 to 5 of (120 - 15(r^2 - 4r cos(theta) + 9)) r dr d(theta)

Simplifying and evaluating this integral gives an average concentration of

Answer: B) 91.25

Solve the Problem. -An Industrial Plant Located in the Center of a Small