Let $x$ be the number of liters of the $30\%$ solution and $y$ be the number of liters of the $50\%$ solution. Since the total volume is $40$ liters,

The amount of pure acid in the final mixture is $42\%$ of $40$, which is

The pure acid from the two starting solutions must add to this amount, so

Substitute $y=40-x$ into the second equation:

Simplify:

Therefore, $16$ liters of the $30\%$ acid solution were used.

First find the total revenue for each day. On Friday, the cafe sold 120 bottles at \1.50$each,sotherevenuewas$120(1.50)=180$dollars.OnSaturday,thecafesold150bottlesat$\$1.80$ each, so the revenue was $150(1.80)=270$ dollars. The increase in revenue was $270-180=90$ dollars. To find the percent increase, divide the increase by the original amount: $\frac{90}{180}=0.5$. Converting to a percent gives $0.5\times 100=50\%$. So the correct answer is $50\%$.

Find the unit price of each bag. The small bag costs $\frac{3.60}{12}=0.30$ dollars per ounce, or $30$ cents per ounce. The large bag costs $\frac{5.40}{20}=0.27$ dollars per ounce, or $27$ cents per ounce. Since $27$ cents is $3$ cents less than $30$ cents, the large bag has the lower cost per ounce. Therefore, the statement that must be true is that the large bag is the better buy because its cost per ounce is $3$ cents less than the small bag's.

First find the bottle's full volume using the formula for the volume of a cylinder: $V=\pi r^{2}h$. Substituting $r=3$ and $h=20$ gives $V\approx 3.14(3^2)(20)=3.14(9)(20)=565.2$ cubic centimeters. Since the bottle is filled to $80\%$ of its capacity, multiply by $0.8$: $0.8(565.2)=452.16$. Because $1$ cubic centimeter equals $1$ milliliter, the bottle contains approximately $452$ milliliters of water.

The recipe uses $3.6$ liters for $24$ servings, so the amount of broth per serving is $\frac{3.6}{24}=0.15$ liter. For $50$ servings, the restaurant needs $50(0.15)=7.5$ liters of broth. Each container holds $750$ milliliters, which is $0.75$ liter. The number of containers needed is $\frac{7.5}{0.75}=10$. Therefore, the least number of containers the restaurant must buy is $10$.