I have build a mathematical model that proportionalizes the weight of a bird to
ID: 1943666 • Letter: I
Question
I have build a mathematical model that proportionalizes the weight of a bird to the length of the legs.
Assume that body density is constant depicted in the model as P.
Assume that we stick this bird in a blender, and it creats a outer layer and an inner layer.
lleg3= inner core (cubed because of volume)
lleg2= outer core (squared because of the surface)
Weight= P + k1 lleg3 = k2 lleg2
Here is my question,
One bird weights 4 1/2 oz with 8 inch long legs, the other weights 64 oz, using the model can you find the length of the legs of the second bird?
P.S.; any critique will help.
Explanation / Answer
I have build a mathematical model that proportionalizes the weight of a bird to the length of the legs.
Assume that body density is constant depicted in the model as P.
Assume that we stick this bird in a blender, and it creats a outer layer and an inner layer.
lleg3= inner core (cubed because of volume)
lleg2= outer core (squared because of the surface)
Weight= P + k1 lleg3 = k2 lleg2
Here is my question,
One bird weights 4 1/2 oz with 8 inch long legs, the other weights 64 oz, using the model can you find the length of the legs of the second bird?
P.S.; any critique will help
-------------------------
Weight= P + k1 lleg3 = k2 lleg2....HOPE YOU MEAN
W=P+K1[L^3]+K2[L^2]...................................1
ASSUMING SO ..
SO WE HAVE 2 CONSTANTS K1 AND K2 WHICH ARE TO BE DETERMINED TO FIT OUR MODEL..
SO WE NEED A MINIMUM OF 2 DATA SETS TO FIND THEM .
BUT IN PRACTICE ,WHEN WE ARE ATTEMPTING TO MODEL SOME PHENOMENA,WE SHOULD
HAVE MANY MORE DATA POINTS TO STATISTICALLY FIND THE BEST FIT.
HERE IN YOUR CASE THERE IS ONLY ONE SET OF DATA POINTS ...VIZ..
One bird weights 4 1/2 oz with 8 inch long legs,
WHICH GIVES ON SUBSTITUTION IN 1
W=4.5=P+K1[8^3]+K2[8^2]=P+512K1+64K2..........................2
FOR THE SECOND BIRD WE HAVE
64=P+K1[L^3]+K2[L^2]........................................3
THERE IS NO WAY WE CAN FIND UNIQIELY THE VALUE OF L FROM THE 2 EQNS. 2 AND 3 ,
WITH 3 UNKNOWNS...K1,K2,...AND L .
AS INFORMED ABOVE YOU SHOULD HAVE A MINIMUM OF TWO DATA SETS AND IN FACT
MANY MORE THAN THAT TO GET A GOOD MODEL ..
SEE EXAMPLE BELOW ..
Y=A+BX+CX^2
AN+B[SIGMA(XI)]+C[SIGMA[XI^2]=[SIGMA(YI)]
A[SIGMA(XI)]+B[SIGMA(XI^2)]+C[SIGMA(XI^3)]=[SIGMA(XI*YI)]
A[SIGMA(XI^2)]+B[SIGMA[(XI^3)]+C[SIGMA(XI^4)]=[SIGMA(XI*XI*YI)]
HENCE IN MATRIX FORM
M*X=K
WHERE
M=
N
S(XI)
S(XI^2)
S(XI)
S(XI^2)
S(XI^3)
S(XI^2)
S(XI^3)
S(XI^4)
X=
A
B
C
K=
S(YI)
S(XI*YI)
S(XI*XI*YI)
SO WE HAVE
N
XI
YI
XI^2
XI^3
XI^4
XI*YI
XI*XI*YI
1
0
1.0
0.0
0.0
0.0
0.0
0.0
2
2
2.0
4.0
8.0
16.0
4.0
8.0
3
3
3.0
9.0
27.0
81.0
9.0
27.0
4
4
5.0
16.0
64.0
256.0
20.0
80.0
SIGMA =
9
11
29
99
353
33
115
M=
4
9
29.0
9
29
99.0
29
99
353.0
K=
11
33
115
X = M INVERSE * K
M INVERSE =
0.991
-0.695
0.114
-0.695
1.298
-0.307
0.114
-0.307
0.080
X = M INVERSE * K=
1.018182
-0.10909
0.272727
HENCE THE REQUIRED L.S.A.IS
Y=
1.0182
+
-0.1
*X+
0.3
*X*X
Y=A+BX+CX^2
AN+B[SIGMA(XI)]+C[SIGMA[XI^2]=[SIGMA(YI)]
A[SIGMA(XI)]+B[SIGMA(XI^2)]+C[SIGMA(XI^3)]=[SIGMA(XI*YI)]
A[SIGMA(XI^2)]+B[SIGMA[(XI^3)]+C[SIGMA(XI^4)]=[SIGMA(XI*XI*YI)]
HENCE IN MATRIX FORM
M*X=K
WHERE
M=
N
S(XI)
S(XI^2)
S(XI)
S(XI^2)
S(XI^3)
S(XI^2)
S(XI^3)
S(XI^4)
X=
A
B
C
K=
S(YI)
S(XI*YI)
S(XI*XI*YI)
SO WE HAVE
N
XI
YI
XI^2
XI^3
XI^4
XI*YI
XI*XI*YI
1
0
1.0
0.0
0.0
0.0
0.0
0.0
2
2
2.0
4.0
8.0
16.0
4.0
8.0
3
3
3.0
9.0
27.0
81.0
9.0
27.0
4
4
5.0
16.0
64.0
256.0
20.0
80.0
SIGMA =
9
11
29
99
353
33
115
M=
4
9
29.0
9
29
99.0
29
99
353.0
K=
11
33
115
X = M INVERSE * K
M INVERSE =
0.991
-0.695
0.114
-0.695
1.298
-0.307
0.114
-0.307
0.080
X = M INVERSE * K=
1.018182
-0.10909
0.272727
HENCE THE REQUIRED L.S.A.IS
Y=
1.0182
+
-0.1
*X+
0.3
*X*X
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