Select the correct answer.
Two hot air balloons are flying above a park. One balloon started at a height of 3,000 feet above the ground and is decreasing in height at a rate of 40 feet per minute. The second balloon is rising at a rate of 50 feet per minute after beginning from a height of 1,200 feet above the ground.
Given that $h$ is the height of the balloons after $m$ minutes, determine which system of equations represents this situation.
A. $h=3,000-40 m$
B. $h=3,000+40 m$ $h=1,200-50 m$
C. $h=3,000 m-40$ $h=1,200 m+50$
D. $m=3,000-40 n$ $m=1,200+50 h
Answer (2)
The height of the first balloon decreases from 3000 feet at a rate of 40 feet per minute, represented by h = 3000 − 40 m .
The height of the second balloon increases from 1200 feet at a rate of 50 feet per minute, represented by h = 1200 + 50 m .
Compare the derived equations with the given options to find the correct system of equations.
Option A, h = 3000 − 40 m , correctly represents the height of the first balloon, so the answer is h = 3 , 000 − 40 m .
Explanation
Problem Analysis Let's analyze the problem. We have two hot air balloons. The first one starts at 3000 feet and descends at 40 feet per minute. The second one starts at 1200 feet and ascends at 50 feet per minute. We need to find the system of equations that represents the height h of each balloon after m minutes.
Equation Setup For the first balloon, the height h decreases from its initial height of 3000 feet by 40 feet each minute. So, the equation for the first balloon is: h = 3000 − 40 m For the second balloon, the height h increases from its initial height of 1200 feet by 50 feet each minute. So, the equation for the second balloon is: h = 1200 + 50 m
Comparing with Options Now, let's compare our equations with the given options: Option A: h = 3000 − 40 m This matches the equation for the first balloon, but we need a system of equations, so we need to check if the second equation is correct. Option B: h = 3000 + 40 m h = 1200 − 50 m These equations are incorrect because the first balloon is descending (so it should be a subtraction) and the second balloon is ascending (so it should be an addition). Option C: h = 3000 m − 40 h = 1200 m + 50 These equations are incorrect because the initial heights are not correctly represented. Option D: m = 3000 − 40 h m = 1200 + 50 h These equations are incorrect because they are solving for m in terms of h , not h in terms of m . Also, the first balloon is descending, so it should be a subtraction, but it's not.
Final Decision Since option A only provides one equation, we need to find the correct system of equations. The correct system should include: h = 3000 − 40 m h = 1200 + 50 m However, none of the options provide the complete correct system. Option A provides the first equation correctly. Let's re-examine the options.
Option A only gives one equation. However, based on our analysis, the first equation should be h = 3000 − 40 m and the second equation should be h = 1200 + 50 m . Since none of the options provide the correct system of equations, let's choose the one that has at least one correct equation. Option A has the equation for the first balloon correct.
Examples
Understanding how to model the height of hot air balloons using linear equations can be applied to various real-world scenarios. For instance, you can model the depreciation of a car's value over time, the growth of a plant, or the balance of a bank account with regular withdrawals or deposits. In each case, identifying the initial value and the rate of change allows you to create a linear equation that predicts the state of the system at any given time. This skill is valuable in making informed decisions and predictions in everyday life.
To correctly represent the situation of two hot air balloons, the equations needed are h = 3000 − 40 m for the first balloon and h = 1200 + 50 m for the second. However, only Option A provides one correct equation. Therefore, the selected option is A: h = 3000 − 40 m .
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