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

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Q. A hot air balloon reaches its maximum cruising height of 1,500ft 1,500 \mathrm{ft} above sea level. Over the next 55 minutes it descends at a constant rate to a new cruising altitude of 1,200ft 1,200 \mathrm{ft} above sea level. If x x represents the time, in minutes, after starting the initial descent, and y y represents the height, in feet, of the hot air balloon, which of the following equations best models the situation for 0≤x≤5 0 \leq x \leq 5 ?\newlineChoose 11 answer:\newline(A) y=1,500−5x y=1,500-5 x \newline(B) y=1,500−60x y=1,500-60 x \newline(C) y=1,500−300x y=1,500-300 x \newline(D) y=1,500−1,200x y=1,500-1,200 x
  1. Determine rate of descent: First, 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}.
  2. Calculate rate per minute: Next, we calculate the rate of descent per minute. Since the descent takes 55 minutes, we divide the total change in altitude by the time, which gives us 300 ft/5 minutes=60 ft per minute.300\,\text{ft} / 5\,\text{minutes} = 60\,\text{ft per minute}.
  3. Create altitude equation: Now, we can create an equation that models the balloon's altitude as a function of time. The initial altitude is 1,500ft1,500\text{ft}, and the balloon descends at a rate of 60ft60\text{ft} per minute. Therefore, the equation is y=1,500−60xy = 1,500 - 60x, where yy is the altitude in feet and xx is the time in minutes.
  4. Validate equation for interval: We must 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 altitude. 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 altitude. This confirms that the equation is correct for the given time interval.

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