Question 778320: The power produced by a wind turbine was tested at different wind speeds and the results tabulated. It was known that there should be a cubic polynomial relationship between wind speed and power. From this data ,estimate the maximum power likely to be produced by this turbine, and the wind speed at which this occurs.
Wind velocity (m/s) 0 3 5 12 20 40 60
Power (KW) 0 1.2 4.5 20.3 41.2 45 33.8
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! The power produced by a wind turbine was tested at different wind speeds and the results tabulated.
It was known that there should be a cubic polynomial relationship between wind speed and power.
From this data ,estimate the maximum power likely to be produced by this turbine, and the wind speed at which this occurs.
:
Wind velocity (m/s) 0 3 5 12 20 40 60
Power (KW) 0 1.2 4.5 20.3 41.2 45 33.8
:
A cubic equation in the a form: ax^3 + bx^2 + cx + d = y
x = 3, y = 1.2
a3^3 + b3^2 + 3c + d = 1.2
27a + 9b + 3c + d = 1.2
:
x = 12, y = 20.3
1728a + 144b + 12c + d = 20.3
:
x = 20, y = 41.2
8000a + 400b + 20c + d = 41.2
:
x = 40, y = 45
64000a + 1600b + 40c + d = 45
:
Construct a matrix
27 9 3 1 1.2
1728 144 12 1 20.3
8000 400 20 1 41.2
64000 1600 40 1 45
Using the matrix feature found
a =-.003
b = .1379
c = .642
d = -1.88
for an equation of:
y = -.003x^3 + .138x^2 + .642x - 1.88
producing a curve of:
:
Max: x = 32.8; y = 61.8, green line at y=62
Maximum power of 62 Kw occurs at a wind-speed of 33 mph
:
this approximates the data, except for 60 mph
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