To solve the problem, we need to find [tex]\( f(g(x)) \)[/tex] and interpret what it represents. Let's start by breaking down the problem into manageable steps.
### Step 1: Understand the Functions
We are given two functions:
[tex]\[ f(x) = 2x^2 + 4 \][/tex]
[tex]\[ g(x) = \sqrt{3x^3} \][/tex]
Here:
- [tex]\( f(x) \)[/tex] represents the amount of money Barrett earns per gallon of ice cream, where [tex]\( x \)[/tex] is the number of gallons.
- [tex]\( g(x) \)[/tex] represents the number of gallons of ice cream Barrett makes per hour, where [tex]\( x \)[/tex] is the number of hours he works.
### Step 2: Compute [tex]\( g(x) \)[/tex]
We first need to determine [tex]\( g(x) \)[/tex] which calculates the number of gallons Barrett makes in a given number of hours.
Let's calculate [tex]\( g(x) \)[/tex] for [tex]\( x = 1 \)[/tex] (assuming Barrett works for 1 hour):
[tex]\[ g(1) = \sqrt{3(1)^3} = \sqrt{3 \cdot 1} = \sqrt{3} \approx 1.732 \][/tex]
Thus, [tex]\( g(1) = \sqrt{3} \approx 1.732 \)[/tex].
### Step 3: Compute [tex]\( f(g(x)) \)[/tex]
Next, we need to find [tex]\( f(g(x)) \)[/tex], which is the amount of money Barrett earns for the number of gallons of ice cream he makes in one hour.
Since [tex]\( g(1) \approx 1.732 \)[/tex], we substitute this value into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(1)) = f(\sqrt{3}) \approx f(1.732) \][/tex]
Using the function [tex]\( f(x) = 2x^2 + 4 \)[/tex]:
[tex]\[ f(1.732) = 2(1.732)^2 + 4 \][/tex]
Calculate [tex]\( (1.732)^2 \)[/tex]:
[tex]\[ (1.732)^2 \approx 2.999 \][/tex]
So, substituting back:
[tex]\[ f(1.732) \approx 2 \cdot 2.999 + 4 = 5.998 + 4 = 9.998 \approx 10 \][/tex]
Thus, [tex]\( f(g(1)) \approx 10 \)[/tex].
### Step 4: Interpretation of [tex]\( f(g(x)) \)[/tex]
The value [tex]\( f(g(x)) \)[/tex] represents the amount of money Barrett earns after working for [tex]\( x \)[/tex] hours. Specifically, [tex]\( f(g(1)) \approx 10 \)[/tex] means that if Barrett works for 1 hour, he will make approximately 1.732 gallons of ice cream and earn [tex]$10.
### Summary
For \( x = 1 \):
\[ g(1) \approx 1.732 \]
\[ f(g(1)) \approx 10 \]
So, the interpretation is that if Barrett works for 1 hour, he makes approximately 1.732 gallons of ice cream and earns $[/tex]10.
