Question

Joelle plans to sell two types of balloons at her charity event: 171717‑inch latex balloons that require 222 cubic feet (\text{ft}^3)(ft 3 )left parenthesis, f, t, start superscript, 3, end superscript, right parenthesis of helium and 181818‑inch mylar balloons that require only 0.5 \,\text{ft}^30.5ft 3 0, point, 5, space, f, t, start superscript, 3, end superscript. She only has access to 1{,}000 \,\text{ft}^31,000ft 3 1, comma, 000, space, f, t, start superscript, 3, end superscript of helium, 15\%15%15, percent of which will be unused due to pressure loss in the tanks. She wants to have at least 500500500 balloons for sale in total. For the number of latex balloons, LLL, and number of mylar balloons, MMM, which of the following systems of inequalities best represents this situation?

Answer 1

The system of equations is:
L+M≥500
2L+0.5M≤850

Let L be the number of latex balloons and M be the number of Mylar balloons. She wants the total number of balloons to be at least 500. "At least" means it could be equal to that number or more; thus

L+M≥500

Since each latex balloon holds 2 ft³ of helium, we represent this with 2L. The 0.5 ft³ of helium in each Mylar balloon is represented with 0.5M. The sum of these, the total amount of helium, can be no more than 85% of 1000 (if 15% is lost, then 100%-15%=85% is kept). 85% of 1000 = 0.85(1000) = 850. This leads to our other inequality,

2L + 0.5M ≤ 850