Word Problem

Question

The Carver Foundation funds three non-profit organizations engaged in alternate-energy research activities. From past data, the proportion of funds spent by each organization in research on solar energy, energy from harnessing the wind, and energy from the motion of ocean tides is given in the accompanying table. 

Proportion of Money Spent
Solar Wind Tides
Organization I 0.6 0.3 0.1
Organization II 0.4 0.3 0.3
Organization III 0.2 0.6 0.2

Find the amount awarded to each organization if the total amount spent by all three organizations on solar, wind, and tidal research is as follows.
(a) $10 million, $10.2 million, and $5.8 million, respectively

Proportion of Money Spent
	Solar	Wind	Tides
Organization I	0.6	0.3	0.1
Organization II	0.4	0.3	0.3
Organization III	0.2	0.6	0.2

Answer 1

 

a=money spent on Organization I

b=money spent on Organization II

c=money spent on Organization III

 

0.6a+0.4b+0.2c = 10,000,000  (1)

0.3a+0.3b+0.6c=10,200,000    (2)

0.1a+0.3b+0.2c=5,800,000      (3)      Multiply this equation by 3: 0.3a+0.9b+0.6c = 17,400,000   (4)

 

Subtract (2) from (4):

 

0.6b = 7,200,000   Divide each side by 0.6:   b= 12,000,000

 

 

Subtract (3) from (1):                                         

 

0.5a+0.1b=4,200,000   Plug in the value of b: 0.5a+0.1*12,000,000 = 4,200,000

                                                                    0.5a+1,200,000         = 4,200,000

                                   Subtract 1,200,000 from each side:

                                                                    0.5a                          = 3,000,000

                                   Divide each side by 0.5: a= 6,000,000

 

Plug in the values of a and b in (1), (2), or (3).

Let's choose (3):          0.1a+0.3b+0.2c                                = 5,800,000    

                                  0.1*6,000,000+0.3*12,000,000+0.2c = 5,800,000 

                                  600,000 + 3,600,000 + 0.2c               = 5,800,000

                                  4,200,000 + 0.2c                               = 5,800,000

                                  Subtract 4,200,000 from each side:

                                  0.2c                                                  = 1,600,000

                                  Divide each side by 0.2:                   c = 8,000,000

 

Answer 2

Let

x = Org I total fund

y = Org II total fund

z = Org III total fund

then

0.6x + 0.4y + 0.2z = 10.0

03.x + 0.3y + 0.6z = 10.2

01.x + 0.3y + 0.2z = 5.8

solve system of simultaneous equations several ways.  

 

One way is to multiply third eqn by 6 and subtract first eqn to get rid of x

1.4y + 1.0z = 24.8

multiply second eqn by 2 and subtract first eqn 

0.2y + 1.0z = 10.4

sutract second from first to get

1.2y = 14.4

y = 12

z = 8

x = 6

 

 

 

 

Answer 3

Cece,

 

This is a system of equations with three variables as follows:

.6x+.4y+.2z=10

.3x+.3y+.6z=10.2

.1x+.3y+.2z=5.8

 

You could manipulate equations back and forth until you get one variable solved, then the other, etc. but I prefer matrices. Set up your first matrix (3x3) A:

.6 .4. .2

.3 .3 .6

.1 .3 .2

 

And the other as a 1x3 matrix (B)

10 10.2 5.8

 

Now we want to find the matrix X such that AX=B. In order to find that we find the inverse of A (A^-1) to cancel out A on the left. Then we multiply BA^-1 to find x.

 

To find the inverse of the 3x3 matrix A, I use minverse in Excel--be sure to highlight a 3x3 space and Control-Shift-Enter)

2 1/3 -3

0 -5/3 5

-1 7/3 -1

 

Then I use mmult in Excel to find x=BA^-1=[14.2 -.13 15.2] (be sure to use Control-Shift-Enter)

 

Since Organization 2 is contributing a negative amount here, we can assume that this is zero. Checking our work:

14.2*.6+15.2*.1=8.52+1.52=10

14.2*.4+15.2*.3=5.68+4.56=10.2

14.2*.2+15.2*.2=2.84+3.04=5.88

 

Hope that helps!