Question

Water coming out of 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 feet per second from 3 seconds to 5 seconds.
B. The water falls down with an average speed of 5 feet per second from 3 seconds to 5 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.

Answer 1

Calculating x=3 and x=5, we get f(x)=10 and 4 respectively by plugging the numbers into the equation. Finding the difference and dividing by 2 (since there are 2 seconds between 3 and 5), we get 3, so the answer is either A or C. Since it doesn't only fall 3 feet, we need to specify that it is 3 feet per second, A is right!

Answer 2

The correct option is A) The water falls down with an average speed of 3 feet per second from 3 seconds to 5 seconds.

Explanation:

Consider the provided function.

`f(x)=-x^2+5x+4` where f(x) represents the height, in feet.

Substitute the value of x = 3 in the provided equation.

`f(3) = -x^2 +5x +4`

`f(3) = -(3)^2 +5(3)+4`

`f(3) = -9+15+4`

`f(3) = 10`

Now, substitute the value of x = 5 in the provided equation.

`f(5) = -x^2 +5x +4`

`f(5) = -5^2 +5(5) +4`

`f(5) = -25+25 +4`

`f(5) = 4`

Now, to find the average rate of change of f(x) from x = 3 to x = 5.

Calculate the difference of f(5) and f(3) and divide it by 2, because the difference between 3 and 5 is 2.

`(f(3)-f(5) )/(2)`

`(10-4)/(2)`

`(6)/(2)=3`

The water falls down with an average speed of 3 feet per second from 3 seconds to 5 seconds.

Thus, the correct option is A) The water falls down with an average speed of 3 feet per second from 3 seconds to 5 seconds.