Water coming out from a fountain is modeled by the function f(x)= -x^2+5x+4 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=3 to x=5 represent?
A. The water falls down with an average speed of 3 ft./s from three seconds to five seconds.
B. The water falls down with an average speed of 5 ft./s from three seconds to five seconds.
C. The water travels an average distance of 3 feet from 3 seconds to 5 seconds.
D. The water travels an average distance of 5 feet from 3 seconds to 5 seconds.
Thank you so much!
To find the average rate of change of a function, we need to calculate the change in its values over a given interval. In this case, we want to find the average rate of change of the function f(x) = -x^2 + 5x + 4 from x = 3 to x = 5.
The average rate of change is the difference in the function's values divided by the difference in the inputs (x-values). Mathematically, it is represented as:
Average Rate of Change = (f(x2) - f(x1)) / (x2 - x1)
Given x1 = 3 and x2 = 5, let's proceed with the calculation.
Step 1: Determine f(x1) and f(x2)
Substitute x1 = 3 and x2 = 5 into f(x) = -x^2 + 5x + 4 to get:
f(3) = -(3)^2 + 5(3) + 4 = -9 + 15 + 4 = 10
f(5) = -(5)^2 + 5(5) + 4 = -25 + 25 + 4 = 4
Step 2: Calculate the average rate of change
Using the formula, we can now find the average rate of change:
Average Rate of Change = (f(5) - f(3)) / (5 - 3) = (4 - 10) / (2) = -6 / 2 = -3
Therefore, the average rate of change of f(x) from x = 3 to x = 5 is -3. This value represents the average decrease in height per second during the time interval from x = 3 to x = 5.
@bonsnian, nevermind I’m a !@#$%^&. !@#$%^& me.
@bonsnian, I used my iPhone calculator (simple calculator)
@bonsnian, -3^2 is NOT -9, it’s 9. -5^2 is NOT -25. You are an idiot. Double check your answers before posting them.
f ( x ) = - x ^ 2 + 5 x + 4
f ( 3 ) = - 3 ^ 2 + 5 * 3 + 4 = - 9 + 15 + 4 = 10
f ( 5 ) = - 5 ^ 2 + 5 * 5 + 4 = - 25 + 25 + 4 = 4
The average rate of change :
[ f ( 5 ) - f ( 3 ) ] / ( 5 - 3 ) =
( 10 - 4 ) / ( 5 - 3 )
= 6 / 2 = 3
Answer A