Water coming out from a fountain is modeled by the function f(x) = −x2 + 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?
The water travels an average distance of 3 feet from 1 second to 4 seconds.
The water travels an average distance of 2 feet from 1 second to 4 seconds.
The water rises up with an average speed of 3 feet per second from 1 second to 4 seconds.
The water rises up with an average speed of 2 feet per second from 1 second to 4 seconds.
f(1) = -1+8+2 = 9
f(4) = -16+32+2 = 18
So, water traveled 9 ft in 3 seconds.
Looks like (C) to me
To find the average rate of change of a function from x = 1 to x = 4, we need to compute the difference in the function values at these two points and divide it by the difference in x-values.
Given the function f(x) = -x^2 + 8x + 2, we need to find f(4) and f(1).
Substituting x = 4 into the function, we get f(4) = -4^2 + 8(4) + 2 = -16 + 32 + 2 = 18.
Substituting x = 1 into the function, we get f(1) = -1^2 + 8(1) + 2 = -1 + 8 + 2 = 9.
Now, we can calculate the average rate of change:
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 represents the water traveling an average distance of 3 feet from 1 second to 4 seconds. Therefore, the correct answer is: The water travels an average distance of 3 feet from 1 second to 4 seconds.