A city planning commission is seeking to identify prime locations for a new zoo and a new hospital. The city's economy is heavily dependent on two industrial plants located relatively near the city center. Both emit particulate matter into the atmosphere, and the resulting air pollution is of concern to the commission and will influence its decisions. The consensus of the commission is that the new zoo should be built in the least polluted area within the city limits, and the new hospital should be built in the least polluted location within two miles of the city center.
The boundaries of the city form a rectangle ten miles from east to west and six miles from north to south. When a coordinate system is chosen with the origin at the center of the rectangle (the city center), Industrial Plant 1 has coordinates (-1, 1) and Industrial Plant 2 has coordinates (1, 0). At a point, (x, y), the concentration of particulate matter (in parts per million) due to emissions from Plant 1 is given by C1 = 200 - 3(d1)2, where d1 is the distance from (x, y) to Plant 1. Similarly, the concentration due to Plant 2 is given by C2 = 200 - 3(d2)2, where d2 is the distance from (x, y) to Plant 2.
(A) Find the point within the city limits that has the greatest concentration of particulate matter.
(B) Find the points on the city boundary that have the greatest and least concentrations of particulate matter.
(C) Find the average concentration of particulate matter throughout the city.
(D) Find the points on the circle of radius two miles, centered at the origin, that have the greatest and least concentrations of particulate matter.
(E) Determine the optimal locations for the city's new zoo and new hospital.
Help what can i do?
The boundaries of the city form a rectangle ten miles from east to west and six miles from north to south. When a coordinate system is chosen with the origin at the center of the rectangle (the city center), Industrial Plant 1 has coordinates (-1, 1) and Industrial Plant 2 has coordinates (1, 0). At a point, (x, y), the concentration of particulate matter (in parts per million) due to emissions from Plant 1 is given by C1 = 200 - 3(d1)2, where d1 is the distance from (x, y) to Plant 1. Similarly, the concentration due to Plant 2 is given by C2 = 200 - 3(d2)2, where d2 is the distance from (x, y) to Plant 2.
(A) Find the point within the city limits that has the greatest concentration of particulate matter.
(B) Find the points on the city boundary that have the greatest and least concentrations of particulate matter.
(C) Find the average concentration of particulate matter throughout the city.
(D) Find the points on the circle of radius two miles, centered at the origin, that have the greatest and least concentrations of particulate matter.
(E) Determine the optimal locations for the city's new zoo and new hospital.
Help what can i do?
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