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
:
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:
{{{ graph( 300, 200, -10, 50, -20, 80, -.003x^3 + .138x^2 + .642x - 1.88, 62) }}}
:
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