An agricultural sprinkler distributes water in a circular pattern of radius 110 ft. It supplies water to a depth of e−r feet per
hour at a distance of r feet from the sprinkler. (Do not substitute numerical values; use variables only.) (a) If 0 < R ≤ 110, what is the total amount of water supplied per hour to the region inside the circle of radi
1 answer:
Answer:
the total amount of water supplier per hour to the region within a circle of radius R=110 ( that is from distance r, 0<r<110)
Step-by-step explanation:
if f(r) describes the water supplied at a distance r , the total amount supplied inside a region that goes from 0 until the circle of radius R, is the sum of all f(r) values from 0 until R, that is the integral value over these limits.
The formula deduction can be found in the attached picture
There is an "r" that multiplies e^-r as result of changing from rectangular coordinates to polar ones.(dx*dy --> r*dr*da)
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Answer:
The distance from the shoreline from P the pipe from the island will reach land in order to minimize the total construction cost is 0 m, or the pipe should be routed to the point P
Step-by-step explanation:
We note that, cost to lay pipe in water = 2.3 × cost to lay pipe on land
Distance of the small resort on the island from point p on the shoreline = 3 miles
Location of the closest source of fresh water is 10 miles along the shoreline from p
∴ The small resort, the point p and the closest source of the fresh water on the shoreline together form a right triangle with sides
Height = 3 miles
Base = 10 miles and
Hypotenuse =
To find the cost of laying the pipeline by the various route, we multiply the water routes by 2.3 and the land routes by 1 and we get
Costs:
Height = Water route = 2.3 × 3 miles = 6.9
Base = Land route = 1 × 10 miles = 10
Hypotenuse = Water route = 2.3 × 10.44 miles = 24.012
Therefore, the two routes are either
10 miles to point p and 3 miles across the water to the resort with a cost = 10 + 6.9 = 16.9 or
Directly to the small resort through 10.44 miles across the water with cost = 24.012
Therefore, to minimize cost, the pipe should come from the resort to P from there to the fresh water source.
That is the distance from the shoreline from P the pipe from the island will reach land in order to minimize the total construction cost = 0 m.