Question

To save on helium costs, a balloon is inflated with both helium and nitrogen gas. Between the two gases, the balloon can be inflated up to $8$ liters in volume. The density of helium is $0.20$ grams per liter, and the density of nitrogen is $1.30$ grams per liter. The balloon must be filled so that the volumetric average density of the balloon is lower than that of air, which has a density of $1.20$ grams per liter. Which of the following system of inequalities best describes how the balloon will be filled, if $x$ represents the number of liters of helium and $y$ represents the number of liters of nitrogen ?

• A. $\{\begin{array}{c}x+y>8\\ 20x+130y>120\end{array}$
• B. $\{\begin{array}{c}x+y=8\\ \frac{0.2x+1.30y}{2}<1.20\end{array}$
• C. $\{\begin{array}{c}x+y\le 8\\ 0.20\left(\frac{x}{x+y}\right)+1.30\left(\frac{y}{x+y}\right)<1.20\end{array}$
• D. $\{\begin{array}{c}x+y\le 8\\ 0.20x+1.30y<1.20\end{array}$

Answer 1

Answer: (C)

$\{\begin{array}{c}x+y\le 8\\ 0.20\left(\frac{x}{x+y}\right)+1.30\left(\frac{y}{x+y}\right)<1.20\end{array}$

A balloon is inflated with both helium and nitrogen gas. Between the two gases, the balloon can be inflated up to $8$ liters in volume.

The density of helium is $0.20$ grams per liter, and the density of nitrogen is $1.30$ grams per liter.

The balloon must be filled so that the voluemetric average density of the balloon is lower than that of air, which has a density of $1.20$ grams per liter.

If $x$ represents the number of liters of helium and $y$ represent the number of liters of nitrogen gas, then

$x+y\le 8$.......(1) as per given

Then volumetric density of helium gas $=$ $0.20\left(\frac{x}{x+y}\right)$

Then volumetric density of nitrogen gas = $1.30\left(\frac{y}{x+y}\right)$

Then $0.20\left(\frac{x}{x+y}\right)+1.30\left(\frac{y}{x+y}\right)$ $<1.20$........(2)