Agebra2 LP prob
j
textbook :
You are planning a dinner of printo beans and brown rice. you want to
consume at least 2100 calories and 44 grams of protein per day, but no
more than 2400 milligrams of sodium and 73 grams of fat. So far today,
you have confusmed 1600 calories, 24 grams of protein, 2370 milligrams
of sodium and 65 grams of fat. Printo beans cost $.57 per cup and
brown rice costs $.78 per cup. How many cups of printo beans and brown
rice should u make to minimize cost while satisfying your nutiritional
requirements??
a table is provided along with the sum:
Contents 1 cup 1cup brown rice ( with salt )
Calories 265 230
Protein ( g) 15 5
Sodium ( mg) 3 10
Fat ( g) 1 1
Any help with this problem would be appreciated
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Chergarj
>consume at least 2100 calories and 44 grams of protein per day, but no
>more than 2400 milligrams of sodium and 73 grams of fat. So far today,
>you have confusmed 1600 calories, 24 grams of protein, 2370 milligrams
>of sodium and 65 grams of fat. Printo beans cost $.57 per cup and
>brown rice costs $.78 per cup. How many cups of printo beans and brown
>rice should u make to minimize cost while satisfying your nutiritional
>requirements??
>
That one reminds me of the discussion in an old intermediate algebra textbook
on linear programming. I never understood that section. Still don't
understand it.
I took your problem, and tried to study it; but the best I could do was put the
equations derived from your problem onto a graphing calculator and come up with
a combination like close to, half cup of rice, and two and half cups of beans.
I wish someone could explain more clearly how "linear programming" works. A
couple of college algebra books which I have seen also have sections on linear
programming. VERY CONFUSING. Generally, in real life, I had used sometimes
systems of two or three equations, did not concern myself much with inequality
aspect of the situations, and obtained realistically very good or excellent
results. Linear programming seems to be something beyond that.
G C
Casey Hawthorne
Which is basically an enhanced matrix representation.
The Simplex algorithm tracks along the "convex polytope" until it
finds a maximum.
There is also an "ellipsoidal" algorithm, which works out from the
interior of the convex polytope - I believe.
--
Regards,
Casey
N. Silver
> You are planning a dinner of printo beans and brown rice.
> you want to consume at least 2100 calories and 44 grams
> of protein per day, but no more than 2400 milligrams of
> sodium and 73 grams of fat. So far today, you have
> consumed 1600 calories, 24 grams of protein, 2370 milligrams
> of sodium and 65 grams of fat. Printo beans cost $.57 per cup
> and brown rice costs $.78 per cup. How many cups of printo
> beans and brown rice should u make to minimize cost while
> satisfying your nutiritional requirements??
> a table is provided along with the sum:
> Contents 1 cup 1cup brown rice ( with salt )
> Calories 265 230
> Protein ( g) 15 5
> Sodium ( mg) 3 10
> Fat ( g) 1 1
I assume you meant to put "pinto beans"
between the cups in the above.
We need 500 calories, 20 grams of protein,
and no more than 30 mg of sodium and 8 grams of fat.
Cost function in pennies, C(p, r) = 57p + 78r
Calorie inequality: 265p + 230r >= 500
Protein inequality: 15p + 5r >= 20
Sodium inequality: 3p + 10r <= 30
Fat inequality: p + r <= 8
We also have: p>=0 and r >=0.
These inequalities together, describe the domain of the
cost function of two variables, C. Since the graph of C
is a plane in 3-space and the domain of C is a polygon
(intersection of lines) in the first quadrant in 2-space,
the minimum of C occurs at a "corner" point of the domain.
If I'm right so far, to approximate the minimum of C,
compare the values:
C(.988, 1.035)
C(1.8867, 0)
C(.370, 2.888)
C(6/7, 50/7)
C(8,0)