A diet requires at least 24 units of vitamin C and at least 26 units of vitamin B complex.Two food supplements are available that provide these nutrients, in the amounts and costs shown in the table
Below.How much of each should be used to minimize the cost?

A) $\quad 3$ grams of $A$ and 9 grams of $B : 33$ cents
B) 1 gram of $A$ and 5 grams of B: 27 cents
C) $\quad 5$ grams of $A$ and 7 grams of B: 39 cents
D) $\quad 7$ grams of $A$ and 5 grams of B: 45 cents
E) 5 grams of $A$ and 2 grams of B: 20 cents

Question

A diet requires at least 24 units of vitamin C and at least 26 units of vitamin B complex.Two food supplements are available that provide these nutrients, in the amounts and costs shown in the table
Below.How much of each should be used to minimize the cost?

A) $\quad 3$ grams of $A$ and 9 grams of $B : 33$ cents 
B) 1 gram of $A$ and 5 grams of B: 27 cents 
C) $\quad 5$ grams of $A$ and 7 grams of B: 39 cents 
D) $\quad 7$ grams of $A$ and 5 grams of B: 45 cents 
E) 5 grams of $A$ and 2 grams of B: 20 cents

Quizplus · Accepted Answer

The problem can be expressed as a linear programming problem. Let $x$ be the amount (in grams) of supplement A and $y$ be the amount (in grams) of supplement B. We want to minimize the cost $0.03x + 0.05y$ subject to the constraints: $$x + 5y \geq 24$$$$3x + 2y \geq 26$$$$x, y \geq 0$$
The first constraint means that the total amount of vitamin C provided by the supplements must be at least 24 units. The second constraint means that the total amount of vitamin B complex provided by the supplements must be at least 26 units. The third constraint just requires that we use non-negative amounts of each supplement.

We can graph the feasible region defined by these constraints:

[asy]
import graph;
size(8cm);
defaultpen(linewidth(0.7));
real[] xtick = {0, 5, 10, 15, 20};
real[] ytick = {0, 5, 10, 15, 20, 25};
real xx(real x, real y) {return x;}
real yy(real x, real y) {return y;}
real[] bl = {0, 0};
real[] br = {8, 0};
real[] tl = {0, 6};
fill((0,2)--(5,2)--(5,7)--(0,11)--cycle, rgb(1, 0.7, 0.7));
fill((5,2)--(10,3)--(8,5)--(5,10)--cycle, rgb(0.7, 0.7, 1));
draw((0,0)--(25/3,0),EndArrow);
draw((0,0)--(0,13),EndArrow);
draw(reflect((0,0),(1,1))*(xscale(0.5)*yscale(0.5)*"$x$"), (25/3,0)--(0,0));
draw((0,13)--(0,0),N);
draw(reflect((0,0),(1,1))*(xscale(0.5)*yscale(0.5)*"$y$"), (0,13)--(0,0), W);
label("$(0,0)$", (0,0), SW);
draw(tl--br,dashed);
label("$(8,0)$", (8,0), S);
draw(bl--(0,11),dashed);
draw((0,0)--(4,4.8),dotted);
draw((0,0)--(25/3,8), dotted);
fill((0,0)--(5,0)--(5,2)--(0,2)--cycle, white);
fill((0,0)--(25/3,0)--(25/3,8)--(0,13)--cycle, white);
draw((0,2)--(5,2), linetype("6 6"));
label("$y \geq 5 - 0.2x$", (2, 2),N);
draw((25/3,8)--(10,3), linetype("6 6"));
label("$y \geq (26-3x)/2$", (16/3, 5), NE);[/asy]

The feasible region is the shaded polygon. Each corner of the polygon corresponds to using one of the supplements alone. It remains to check each corner to see which minimizes the cost $0.03x+0.05y$.

Corner A: $(0, \frac{26}{2})$. Cost is $0.05(\frac{26}{2}) = 0.65$.

Corner B: $(\frac{24}{5}, 0)$. Cost is $0.03(\frac{24}{5}) = 0.144$.

Corner C: $(4, 3)$. Cost is $0.03(4) + 0.05(3) = 0.27$.

Corner D: $(\frac{26}{3}, 0)$. Cost is $0.03(\frac{26}{3}) = 0.26$.

Corner E: $(5, 2)$. Cost is $0.03(5) + 0.05(2) = 0.25$.

The minimum cost is achieved at corner E, where 5 grams of supplement A and 2 grams of supplement B are used, costing 20 cents. Therefore, the answer is $\boxed{	extbf{(E)}}$.

A Diet Requires at Least 24 Units of Vitamin C 3\quad 33

A Diet Requires at Least 24 Units of Vitamin C $\quad 3$