An alcohol solution consists of 7 gallons of pure water and x gallons of alcohol, x>0 . Let f(x) be the ratio of the volume of alcohol to the total volume of liquid.

Question

An alcohol solution consists of 7 gallons of pure water and x gallons of alcohol, x>0. Let f(x) be the ratio of the volume of alcohol to the total volume of liquid.

(Note that f(x) is the concentration of the alcohol in the solution.)

(a) Find a formula for  f(x)=

(b) Evaluate  f(5)=

 .

(c) What is the zero of f(x)?  x=

(d) What is the long-term behavior of f(x))?  y=

Answer 1

f(x) = x/(x+7)

f(5) = 5/12

x=0 is the zero

end behavior: f(x) approaches 1, as x approaches infinity

y=1 is the horizontal asymptote

f(x)=y=x/(x+7) = 1 -7/(x+7)

Answer 2

Adri, fyi, I got the same answer as Mark already did. Here are the steps I took with some explanation:

(a) Find a formula for f(x):

The function f(x) represents the ratio of the volume of alcohol to the total volume of the liquid. The total volume of the liquid is 7 + x (7 gallons of water and x gallons of alcohol).

Therefore, the formula for f(x) is: f(x) = x / (7 + x)

(b) Evaluate f(5): To find f(5), simply substitute x = 5 into the formula:

f(5) = 5 / (7 + 5) = 5 / 12

(c) What is the zero of f(x)?

The zero of f(x) is the value of x that makes f(x) = 0.

Since the numerator is x, the only value that makes f(x) zero is x = 0.

(d) What is the long-term behavior of f(x)?

As x approaches infinity, the term x becomes much larger compared to the constant 7.

The function f(x) will approach 1 but never reach it.

Therefore, the long-term behavior is y = 1.

Thank you,

Benjamin M.