Please help me with the following mathematical question.

Question

The yield of zucchini Z kg/hectare, over a period of x hours of sunshine and y mm of rainfall is shown in the following statistics:

Z= 23,28,30
X= 30,50,40
Y= 320,360,400

a) Form a linear model in the form z=a+bx+cy for this data.
b) How many kilograms per hectare of zucchini would the model predict for 40 hours of sunshine and only 200mm of rain?

 

Thanks so much

Answer 1

Hi Jack,

You need to use a matrix to solve this problem three unknown with three equation or brute force.

(A) Find a linear model

a + bX +cY = Z

Z= 23,28,30
X= 30,50,40
Y= 320,360,400

 

a + 30b + 320c = 23     --------(1)

a + 50b + 360c = 28     --------(2)

a + 40b + 400c = 30     --------(3)

 

Substract (1) and (2) from (3) to eliminate 'a' because the coefficient are the same

(3) - (1)

a + 40b + 400c - (a + 30b + 320c) =30-23

10b + 80c = 7              --------(4)

(3) - (2)

a + 40b + 400c - (a + 50b + 360c) = 30-28

-10b + 40c = 2             --------(5)

Add (4) and (5) to eliminate 'b' because the coefficient are opposite in sign

120c = 9

     c = 9/120 = 3/40 

Plug value of 'c' into either (4) or (5) (you will get the same value for b)

-10b + 40c = 2

-10b +40(3/40) = 2

-10b + 3 = 2

     -10b = 1

          b = -1/10

Plug 'b' and 'c' into (1)

a + 30b + 320c = 23

a + 30(-1/10) + 320(3/40) = 23

a  - 3 +24 = 23

     a + 21 =23

             a = 2

Plug the values of 'a', 'b', and 'c'

Z = a + bX +cY 

Z = 2 -X/10 + 3Y/40

(B) X = 40 hrs sunshine, Y = 200mm rain

Z = 2 -X/10 +3Y/40

   = 2 -40/10 + 3(200)/40

   = 2 - 4 +15 = 13

Z =13 kg//hectare