Question

Water coming out from a fountain is modeled by the function:
f(x)=−x2+6x+6
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 = 2 to x = 5 represent?

a.The water travels an average distance of 4 feet from 2 seconds to 5 seconds.

b.The water travels an average distance of 1 foot from 2 seconds to 5 seconds.

c.The water falls down with an average speed of 2 feet per second from 2 seconds to 5 seconds.

d. The water falls down with an average speed of 1 foot per second from 2 seconds to 5 seconds.

Answer 1

Average rate of change from x = 2 to x = 5 is (f(5) - f(2))/(5 - 2) = ((-(5)^2+6(5)+6) - (-(2)^2+6(2)+6))/3 = ((-25+30+6) - (-4+12+6))/3 = (11 - 14)/3 = -3/3 = -1

Therefore, the water falls down with an average speed of 1 foot per second from 2 seconds to 5 seconds.

Answer 2

Average speed = change in position / change in time.
m = (f( b ) - f ( a )) / ( b - a )
a = 2,  b = 5
f ( a ) = f ( 2 ) = - 4 + 12 + 6 = 14
f ( b ) = f ( 5 ) = - 25 + 30 + 6 = 11
m = (11-14) / (5-2) = -3/3 = -1
Answer: D ) The water falls down with an average speed of 1 foot per second from 2 seconds to 5 seconds.