Question

a wholesaler uses the function f(x)=9\sqrt{x} to determine the cost, in dollars, to buy x gallons of lemonade. customers can buy partial gallons of lemonade. find and interpret the given function values and determine an appropriate domain for the function.
round your answers to the nearest cent.
answer
f(0)=□, meaning the cost of buying □ gallons of lemonade would be $□. this interpretation - in the context of the problem.
f(10)=□, meaning the cost of buying □ gallons of lemonade would be $□. this interpretation - in the context of the problem.
f(67.5)=□, meaning the cost of buying □ gallons of lemonade would be $□. this interpretation - in the context of the problem.
based on the observations above, it is clear that an appropriate domain for the function is
all real numbers
non - negative real numbers
real numbers in a\leq x\leq b
integers
whole numbers {0, 1, 2, ...}
positive integers {1, 2, 3, ...}
integers in a\leq x\leq b

Explanation:

Step1: Calculate \(f(0)\)

Substitute \(x = 0\) into \(f(x)=9\sqrt{x}\). Since \(\sqrt{0}=0\), then \(f(0)=9\times\sqrt{0}=0\). This means the cost of buying 0 gallons of lemonade is $0. This interpretation is valid in the context of the problem as it represents no - purchase.

Step2: Calculate \(f(10)\)

Substitute \(x = 10\) into \(f(x)=9\sqrt{x}\). So \(f(10)=9\times\sqrt{10}\approx9\times3.162 = 28.46\). This means the cost of buying 10 gallons of lemonade is approximately $28.46. This interpretation is valid as it represents the cost for a non - zero quantity of lemonade.

Step3: Calculate \(f(67.5)\)

Substitute \(x = 67.5\) into \(f(x)=9\sqrt{x}\). First, \(\sqrt{67.5}\approx8.216\), then \(f(67.5)=9\times8.216 = 73.94\). This means the cost of buying 67.5 gallons of lemonade is approximately $73.94. This interpretation is valid as it represents the cost for a non - integer quantity of lemonade.

Step4: Determine the domain

Since \(x\) represents the number of gallons of lemonade and one can buy 0 or positive amounts (including non - integer amounts as partial gallons are allowed), the appropriate domain for the function is non - negative real numbers.

Answer:

\(f(0)=0\), meaning the cost of buying 0 gallons of lemonade would be $0. This interpretation is valid in the context of the problem.
\(f(10)\approx28.46\), meaning the cost of buying 10 gallons of lemonade would be $28.46. This interpretation is valid in the context of the problem.
\(f(67.5)\approx73.94\), meaning the cost of buying 67.5 gallons of lemonade would be $73.94. This interpretation is valid in the context of the problem.
An appropriate domain for the function is non - negative real numbers.