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A hot air balloon reaches its maximum cruising height of 1,500ft above sea level. Over the next 5 minutes it descends at a constant rate to a new cruising altitude of 1,200ft above sea level. If x represents the time, in minutes, after starting the initial descent, and y represents the height, in feet, of the hot air balloon, which of the following equations best models the situation for 0 <= x <= 5 ? Choose 1 answer: (A) y=1,500-5x (B) y=1,500-60 x (C) y=1,500-300 x (D) y=1,500-1,200 x

A hot air balloon reaches its maximum cruising height of 1,500ft1,500\text{ft} above sea level. Over the next 55 minutes it descends at a constant rate to a new cruising altitude of 1,200ft1,200\text{ft} above sea level. If xx represents the time, in minutes, after starting the initial descent, and yy represents the height, in feet, of the hot air balloon, which of the following equations best models the situation for 0≤x≤50 \leq x \leq 5?\newlineChoose 11 answer:\newline(A) y=1,500−5xy=1,500-5x\newline(B) y=1,500−60xy=1,500-60x\newline(C) y=1,500−300xy=1,500-300x\newline(D) y=1,500−1,200xy=1,500-1,200x

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Q. A hot air balloon reaches its maximum cruising height of 1,500ft1,500\text{ft} above sea level. Over the next 55 minutes it descends at a constant rate to a new cruising altitude of 1,200ft1,200\text{ft} above sea level. If xx represents the time, in minutes, after starting the initial descent, and yy represents the height, in feet, of the hot air balloon, which of the following equations best models the situation for 0≤x≤50 \leq x \leq 5?\newlineChoose 11 answer:\newline(A) y=1,500−5xy=1,500-5x\newline(B) y=1,500−60xy=1,500-60x\newline(C) y=1,500−300xy=1,500-300x\newline(D) y=1,500−1,200xy=1,500-1,200x
  1. Determine Rate of Descent: We need to determine the rate of descent of the hot air balloon. The balloon descends from 1,500ft1,500\text{ft} to 1,200ft1,200\text{ft}, which is a change in altitude of 1,500ft−1,200ft=300ft1,500\text{ft} - 1,200\text{ft} = 300\text{ft}. This descent happens over 55 minutes.
  2. Calculate Rate of Descent per Minute: To find the rate of descent per minute, we divide the total change in altitude by the total time taken. The rate of descent is 300 ft/5 minutes=60 ft per minute.300\,\text{ft} / 5\,\text{minutes} = 60\,\text{ft per minute}.
  3. Create Height Equation: Now we can create an equation that models the height of the hot air balloon over time. Since the balloon starts at 1,500ft1,500\text{ft} and descends at a rate of 60ft60\text{ft} per minute, the equation is y=1,500−60xy = 1,500 - 60x, where yy is the height in feet and xx is the time in minutes.
  4. Validate Equation for Time Interval: We need to ensure that the equation is valid for the given time interval, which is from 00 to 55 minutes. Plugging in x=0x = 0, we get y=1,500−60(0)=1,500fty = 1,500 - 60(0) = 1,500\text{ft}, which is the initial height. Plugging in x=5x = 5, we get y=1,500−60(5)=1,500−300=1,200fty = 1,500 - 60(5) = 1,500 - 300 = 1,200\text{ft}, which is the final height. The equation is valid for the given time interval.

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