Water coming out from a fountain is modeled by the function f(x) = -x^2 + 8x + 2 where f(x) represents the height, in feet, of the water from the fountain at different times x, in seconds.
What does the average rate of change of f(x) from x = 1 to x = 4 represent?
A. The water travels an average distance of 3 feet from 1 second to 4 seconds.
B. The water travels an average distance of 2 feet from 1 second to 4 seconds.
C.The water rises up an average distance of 3 feet from 1 second to 4 seconds.
D.The water rises up an average distance of 2 feet from 1 second to 4 seconds.
I think the answer is A.
f(1) = -1 + 8 + 2 = 9
f(4) = -16 + 32 + 2 = 18
avg rate of change = (18-9)/(4-1) = 3 ft/s
this is a rate, you matches it up with A, which states a distance.
You need a rate or speed, so it must be C
To find the average rate of change of a function from one point to another, we use the formula:
Average rate of change = (f(b) - f(a))/(b - a)
In this case, we need to find the average rate of change of the function f(x) = -x^2 + 8x + 2 from x = 1 to x = 4.
Let's calculate it:
f(b) = f(4) = -4^2 + 8 * 4 + 2 = -16 + 32 + 2 = 18
f(a) = f(1) = -1^2 + 8 * 1 + 2 = -1 + 8 + 2 = 9
b - a = 4 - 1 = 3
Now plug these values into the formula:
Average rate of change = (f(4) - f(1))/(4 - 1) = (18 - 9)/3 = 9/3 = 3
Therefore, the average rate of change of f(x) from x = 1 to x = 4 is 3.
Now let's analyze the answer choices:
A. The water travels an average distance of 3 feet from 1 second to 4 seconds. - This statement is not true because the average rate of change represents the change in height, not distance.
B. The water travels an average distance of 2 feet from 1 second to 4 seconds. - This statement is not supported by the calculation we made.
C. The water rises up an average distance of 3 feet from 1 second to 4 seconds. - This statement is not true since the height of the water is decreasing according to the given function.
D. The water rises up an average distance of 2 feet from 1 second to 4 seconds. - This statement is not supported by the calculation we made.
Therefore, the correct answer is none of the given choices.
The average rate of change of a function from x = 1 to x = 4 represents the average slope of the function over that interval. In this case, the function represents the height of the water from the fountain at different times.
To find the average rate of change, we need to calculate the slope between the two points.
f(4) - f(1) = (-4^2 + 8(4) + 2) - (-1^2 + 8(1) + 2)
= (-16 + 32 + 2) - (-1 + 8 + 2)
= (18) - (9)
= 9
x(4) - x(1) = (4 - 1)
= 3
Therefore, the average rate of change is 9/3 = 3.
This means that on average, the height of the water from the fountain increases by 3 feet from 1 second to 4 seconds.
So, the correct answer is C. The water rises up an average distance of 3 feet from 1 second to 4 seconds.