I need help solving this

Question

The number of guests g in a water park t hours after 9 am is given by g(t) = –32t2 + 224t + 263 for 0 ≤ t ≤ 8. (t = 0 corresponds to 9 am)
Find and interpret the average rate of change of g over the interval {t | 4 ≤ t ≤ 7} = [4, 7].

Answer 1

Whenever you see the words "average rate of change" or any sort of rate of change, think of a ratio or fraction. For example, when we drive, we use the units "miles per hour" which means miles/hours.

So to solve this problem, we need to find a fraction. How about a slope? Change in y/change in x is a fraction.

Plug in 4 for t and find the corresponding g, and plug in 7 for t and find the corresponding g. Then you have 2 points that you can find the slope between (4, g(4)) and (7, g(7)).

The formula for slope is (y₁-y₂)/(x₁-x₂) or in this case (g(4)-g(7))/(4-7).

Answer 2

Hi Jallisa, the average rate of change is given by:

 

ARC_a,b = g(b) - g(a) / b - a

 

Here, a = 4 and b = 7

 

g(7) = -32 * 7² + 224 * 7 + 263, or 263

 

g(4) = -32 * 7² + 224 * 7 + 264, or 647

 

Substituting gives:

 

ARC_4,7 = (263 - 647) / (7 - 4)

 

           = -384 / 3

 

           = -128

 

The average rate of change for the interval [4, 7] is -128.

 

This implies the number of guest decrease as the day wears on.

 

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