Question

a wholesaler uses the function ( f(x)=8sqrt{x} ) to determine the cost, in dollars, to buy ( x ) gallons of lemonade. find and interpret the given function values and determine an appropriate domain for the function. customers can buy partial gallons of lemonade. round your answers to the nearest cent.
answer attempt 1 out of 2
( f(0)=square ), meaning the cost of buying ( square ) gallons of lemonade would be ( $ square ). this interpretation ( vee ) in the context of the problem.
( f(20)=square ), meaning the cost of buying ( square ) gallons of lemonade would be ( $ square ). this interpretation ( vee ) in the context of the problem.
( f(24.5)=square ), meaning the cost of buying ( square ) gallons of lemonade would be ( $ square ). this interpretation ( vee ) in the context of the problem.
based on the observations above, it is clear that an appropriate domain for the function is

Explanation:

Step1: Calculate \(f(0)\)

Given the function \(f(x) = 8\sqrt{x}\). Substitute \(x = 0\) into the function:
\(f(0)=8\sqrt{0}\)
Since \(\sqrt{0}=0\), then \(f(0)=8\times0 = 0\). This means the cost of buying \(0\) gallons of lemonade is \(\$0\).

Step2: Calculate \(f(20)\)

Substitute \(x = 20\) into the function \(f(x)=8\sqrt{x}\):
\(f(20)=8\sqrt{20}\)
Simplify \(\sqrt{20}=\sqrt{4\times5}=2\sqrt{5}\approx2\times2.236 = 4.472\)
Then \(f(20)=8\times4.472=35.776\approx35.78\). So the cost of buying \(20\) gallons of lemonade is \(\$35.78\).

Step3: Calculate \(f(24.5)\)

Substitute \(x = 24.5\) into the function \(f(x)=8\sqrt{x}\):
\(f(24.5)=8\sqrt{24.5}\)
\(\sqrt{24.5}=\sqrt{\frac{49}{2}}=\frac{7}{\sqrt{2}}\approx\frac{7}{1.414}\approx4.95\)
\(f(24.5)=8\times4.95 = 39.60\). Thus the cost of buying \(24.5\) gallons of lemonade is \(\$39.60\).

Answer:

• \(f(0) = 0\), meaning the cost of buying \(0\) gallons of lemonade would be \(\$0\).
• \(f(20)=35.78\), meaning the cost of buying \(20\) gallons of lemonade would be \(\$35.78\).
• \(f(24.5)=39.60\), meaning the cost of buying \(24.5\) gallons of lemonade would be \(\$39.60\).