in exercises 21 and 22, simplify the complex fraction.
21. $\dfrac{\left(\dfrac{1}{x} + \dfrac{1}{2}\
ight)}{\left(\dfrac{1}{x^2} + \dfrac{1}{2}\
ight)}
22. \(\dfrac{\left(\dfrac{x^2 + 2x - 15}{2x^3 - x^2}\
ight)}{\left(\dfrac{x - 3}{4x^2 - 2x}\
ight)}
application
23. you open a floral shop with a setup cost of $32,000. the cost of creating one dozen floral arrangements is $148.
(a) write the total cost \(c$ as a function of $x$, the number of dozens of floral arrangements created.
(b) write the average cost per dozen $\overline{c} = c/x$ as a function of $x$, the number of dozens of floral arrangements created.
(c) determine the domain of the function in part (b).
(d) find the value of $\overline{c}(800)$.

Question

Accepted Answer

### Exercise 21: Simplify the Complex Fraction
# Explanation:
## Step 1: Simplify the numerator and denominator separately.
- Numerator: $\frac{1}{x} + \frac{1}{2} = \frac{2 + x}{2x}$ (Find a common denominator, which is $2x$, then add the fractions: $\frac{2}{2x} + \frac{x}{2x} = \frac{2 + x}{2x}$)
- Denominator: $\frac{1}{x^2} + \frac{1}{2} = \frac{2 + x^2}{2x^2}$ (Find a common denominator, which is $2x^2$, then add the fractions: $\frac{2}{2x^2} + \frac{x^2}{2x^2} = \frac{2 + x^2}{2x^2}$)

## Step 2: Divide the simplified numerator by the simplified denominator.
Dividing by a fraction is the same as multiplying by its reciprocal:
$\frac{\frac{2 + x}{2x}}{\frac{2 + x^2}{2x^2}} = \frac{2 + x}{2x} 	imes \frac{2x^2}{2 + x^2}$

## Step 3: Simplify the expression.
Cancel out the common factors:
The $2$ in the numerator and denominator cancels. One $x$ from $2x^2$ cancels with $2x$, leaving $x$ in the numerator.
So we get $\frac{(x + 2) 	imes x}{2 + x^2} = \frac{x(x + 2)}{x^2 + 2}$

# Answer:
$\frac{x(x + 2)}{x^2 + 2}$ (or $\frac{x^2 + 2x}{x^2 + 2}$)

### Exercise 22: Simplify the Complex Fraction
# Explanation:
## Step 1: Factor all the polynomials.
- Numerator of the main fraction: $\frac{x^2 + 2x - 15}{2x^3 - x^2}$
  - Factor the numerator: $x^2 + 2x - 15 = (x + 5)(x - 3)$ (since $5 	imes (-3) = -15$ and $5 + (-3) = 2$)
  - Factor the denominator: $2x^3 - x^2 = x^2(2x - 1)$ (factor out $x^2$)
  - So the numerator of the main fraction becomes $\frac{(x + 5)(x - 3)}{x^2(2x - 1)}$
- Denominator of the main fraction: $\frac{x - 3}{4x^2 - 2x}$
  - Factor the denominator: $4x^2 - 2x = 2x(2x - 1)$ (factor out $2x$)
  - So the denominator of the main fraction becomes $\frac{x - 3}{2x(2x - 1)}$

## Step 2: Divide the simplified numerator by the simplified denominator.
Dividing by a fraction is the same as multiplying by its reciprocal:
$\frac{\frac{(x + 5)(x - 3)}{x^2(2x - 1)}}{\frac{x - 3}{2x(2x - 1)}} = \frac{(x + 5)(x - 3)}{x^2(2x - 1)} 	imes \frac{2x(2x - 1)}{x - 3}$

## Step 3: Simplify the expression.
Cancel out the common factors:
- $(x - 3)$ in the numerator and denominator cancels.
- $(2x - 1)$ in the numerator and denominator cancels.
- $2x$ in the numerator and $x^2$ in the denominator: one $x$ cancels, leaving $x$ in the denominator and $2$ in the numerator.
- So we have $\frac{(x + 5) 	imes 2}{x} = \frac{2(x + 5)}{x}$ (or $\frac{2x + 10}{x}$)

# Answer:
$\frac{2(x + 5)}{x}$ (or $\frac{2x + 10}{x}$)

### Exercise 23: Cost Function and Average Cost
#### Part (a): Total Cost Function
# Explanation:
The total cost $C$ is the sum of the fixed setup cost and the variable cost. The setup cost is $\$32,000$, and the variable cost is $\$148$ per dozen floral arrangements. If $x$ is the number of dozens, then the variable cost is $148x$. So the total cost function is $C(x) = 32000 + 148x$.

# Answer:
$C(x) = 148x + 32000$

#### Part (b): Average Cost Function
# Explanation:
The average cost per dozen $\overline{C}$ is the total cost $C$ divided by the number of dozens $x$. From part (a), $C(x) = 148x + 32000$, so $\overline{C}(x) = \frac{C(x)}{x} = \frac{148x + 32000}{x}$. We can simplify this by splitting the fraction: $\overline{C}(x) = 148 + \frac{32000}{x}$.

# Answer:
$\overline{C}(x) = \frac{148x + 32000}{x}$ (or $\overline{C}(x) = 148 + \frac{32000}{x}$)

#### Part (c): Domain of the Average Cost Function
# Explanation:
The domain of a function is the set of all possible input values (in this case, $x$) for which the function is defined. For the average cost function $\overline{C}(x) = \frac{148x + 32000}{x}$, we need to consider two things:
1. The number of dozens $x$ cannot be zero (since we can't divide by zero).
2. In the context of the problem, $x$ represents the number of dozens of floral arrangements, so it must be a positive real number (you can't have a negative number of dozens, and it can be a non - integer in theory, but in practical terms, it's a positive real number).

So the domain is all positive real numbers, or in interval notation, $(0, \infty)$. If we consider only whole numbers of dozens (since you can't have a fraction of a dozen in terms of production in some cases), the domain would be all positive integers. But usually, for cost functions, we consider $x > 0$ (real numbers) as the domain.

# Answer:
If we consider $x$ as a positive real number (for the mathematical function), the domain is $(0, \infty)$. If we consider $x$ as the number of dozens (positive integers), the domain is all positive integers. (Commonly, for the function $\overline{C}(x)=\frac{148x + 32000}{x}$, the domain is $x>0$, $x\in\mathbb{R}$)

#### Part (d): Find $\overline{C}(800)$
# Explanation:
To find $\overline{C}(800)$, we substitute $x = 800$ into the average cost function $\overline{C}(x)=148+\frac{32000}{x}$.

## Step 1: Substitute $x = 800$ into the function.
$\overline{C}(800)=148+\frac{32000}{800}$

## Step 2: Calculate the fraction.
$\frac{32000}{800}=40$

## Step 3: Add the two terms.
$148 + 40=188$

# Answer:
$\overline{C}(800) = 188$ (The average cost per dozen when 800 dozens are produced is $\$188$)

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QUESTION IMAGE

Question

Explanation:

Exercise 21: Simplify the Complex Fraction

Step 1: Simplify the numerator and denominator separately.

Step 2: Divide the simplified numerator by the simplified denominator.

Step 3: Simplify the expression.

Step 1: Factor all the polynomials.

Step 2: Divide the simplified numerator by the simplified denominator.

Step 3: Simplify the expression.

Answer:

Exercise 22: Simplify the Complex Fraction