Question

a small organic food store makes two types of fruit smoothies: tropical and sport. each tropical smoothie uses 9 oz of orange juice and 6 oz of milk. each sport smoothie uses 3 oz of orange juice and 3 oz of milk. the store buys the orange juice in a 210 - oz container and milk in a 150 - oz container. the store sells the tropical smoothies for $4 and sells the sport smoothie for $3. what is the maximum in smoothie sales the store can make from 1 container each of orange juice and milk?
$100
$110
$150
$280

Explanation:

Step1: Define variables

Let $x$ be the number of tropical smoothies and $y$ be the number of sport smoothies.

Step2: Set up constraints

For orange - juice: $9x + 3y\leq210$, which simplifies to $3x + y\leq70$. For milk: $6x+3y\leq150$, which simplifies to $2x + y\leq50$. Also, $x\geq0,y\geq0$.

Step3: Write the objective function

The profit function $P = 4x + 3y$.

Step4: Solve the system of inequalities for corner - points

Solve the system

$$\begin{cases}3x + y=70\\2x + y=50\end{cases}$$

. Subtract the second equation from the first: $(3x + y)-(2x + y)=70 - 50$, so $x = 20$. Substitute $x = 20$ into $2x + y=50$, we get $y=10$. The corner - points of the feasible region are $(0,0),(0, 50),(70/3,0),(20,10)$.

Step5: Evaluate the objective function at corner - points

$P(0,0)=4\times0 + 3\times0=0$. $P(0,50)=4\times0+3\times50 = 150$. $P(\frac{70}{3},0)=4\times\frac{70}{3}+3\times0=\frac{280}{3}\approx93.33$. $P(20,10)=4\times20+3\times10=80 + 30=110$.

Answer:

$110$, so the answer is $\$110$.