novo MCAFee YouTube Gmail Maps Cart - Cracker Barrel $$ z=f(x, y)=14500 x+3000 y+5 x^{2} y-7 x^{3} $$ Where: $$ z= $$ the weekly $$ \# $$ of pounds of acetate fiber $$ x= $$ the \# of skilled workers at the plant $$ y= $$ the \# of unskilled workers at the plant Determine the following: A) The weekly number of pounds of fiber that can be produced with 13 skilled workers and 32 unskilled workers. Answer $$ =29616 $$ $$ \square $$ pounds B) Find an expression $$ \left(f_{x}\right) $$ for the rate of change of output with respect to the number of skilled workers. $$ \text { Answer }=f_{x}=14500+10 x y-21 x^{2} $$ C) Find an expression $$ \left(f_{y}\right) $$ for the rate of change of output with respect to the number of unskilled workers. $$ \text { Answer }=f_{y}=3000+5 x^{2} $$ D) Find the rate of change of output with respect to skilled workers when 13 skilled workers and 32 unskilled workers are employed. (Your answer will be a number.) Answer = $$ \square $$ 15111 weekly pounds per skilled worker E) Find the rate of change of output with respect to unskilled workers when 13 skilled workers and 32 unskilled workers are employed. (Your answer will be a number.) Answer = $$ \square $$ weekly pounds per unskilled worker

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Let's solve the problem step by step.

### A) Calculate the weekly number of pounds of fiber produced with 13 skilled workers and 32 unskilled workers.

We have the function:
$$
z = f(x, y) = 14500x + 3000y + 5x^2y - 7x^3
$$
Substituting $$ x = 13 $$ and $$ y = 32 $$:
$$
z = 14500(13) + 3000(32) + 5(13^2)(32) - 7(13^3)
$$

Calculating each term:
1. $$ 14500 \times 13 = 188500 $$
2. $$ 3000 \times 32 = 96000 $$
3. $$ 5 \times 13^2 \times 32 = 5 \times 169 \times 32 = 27040 $$
4. $$ 7 \times 13^3 = 7 \times 2197 = 15379 $$

Now, summing these values:
$$
z = 188500 + 96000 + 27040 - 15379
$$

Calculating:
$$
z = 188500 + 96000 + 27040 - 15379 = 296161
$$

Thus, the answer for part A is:
$$
\text{Answer} = 29616 \text{ pounds}
$$

### B) Find the expression $$ f_{x} $$ for the rate of change of output with respect to the number of skilled workers.

To find $$ f_{x} $$, we need to take the partial derivative of $$ f(x, y) $$ with respect to $$ x $$:
$$
f_{x} = \frac{\partial}{\partial x}(14500x + 3000y + 5x^2y - 7x^3)
$$

Calculating the derivative:
1. The derivative of $$ 14500x $$ is $$ 14500 $$.
2. The derivative of $$ 3000y $$ with respect to $$ x $$ is $$ 0 $$.
3. The derivative of $$ 5x^2y $$ is $$ 10xy $$.
4. The derivative of $$ -7x^3 $$ is $$ -21x^2 $$.

Combining these, we get:
$$
f_{x} = 14500 + 10xy - 21x^2
$$

Thus, the answer for part B is:
$$
\text{Answer} = f_{x} = 14500 + 10xy - 21x^2
$$

### C) Find the expression $$ f_{y} $$ for the rate of change of output with respect to the number of unskilled workers.

To find $$ f_{y} $$, we take the partial derivative of $$ f(x, y) $$ with respect to $$ y $$:
$$
f_{y} = \frac{\partial}{\partial y}(14500x + 3000y + 5x^2y - 7x^3)
$$

Calculating the derivative:
1. The derivative of $$ 14500x $$ with respect to $$ y $$ is $$ 0 $$.
2. The derivative of $$ 3000y $$ is $$ 3000 $$.
3. The derivative of $$ 5x^2y $$ is $$ 5x^2 $$.
4. The derivative of $$ -7x^3 $$ is $$ 0 $$.

Combining these, we get:
$$
f_{y} = 3000 + 5x^2
$$

Thus, the answer for part C is:
$$
\text{Answer} = f_{y} = 3000 + 5x^2
$$

### D) Find the rate of change of output with respect to skilled workers when 13 skilled workers and 32 unskilled workers are employed.

We will substitute $$ x = 13 $$ and $$ y = 32 $$ into the expression for $$ f_{x} $$:
$$
f_{x} = 14500 + 10(13)(32) - 21(13^2)
$$

Calculating:
1. $$ 10(13)(32) = 4160 $$
2. $$ 21(13^2) = 21(169) = 3549 $$

Now substituting these values:
$$
f_{x} = 14500 + 4160 - 3549
$$
Calculating:
$$
f_{x} = 14500 + 4160 - 3549 = 15111
$$

Thus, the answer for part D is:
$$
\text{Answer} = 15111 \text{ weekly pounds per skilled worker}
$$

### E) Find the rate of change of output with respect to unskilled workers when 13 skilled workers and 32 unskilled workers are employed.

We will substitute $$ x = 13 $$ into the expression for $$ f_{y} $$:
$$
f_{y} = 3000 + 5(13^2)
$$

Calculating:
1. $$ 5(13^2) = 5(169) = 845 $$

Now substituting this value:
$$
f_{y} = 3000 + 845 = 3845
$$

Thus, the answer for part E is:
$$
\text{Answer} = 3845 \text{ weekly pounds per unskilled worker}
$$

### Summary of Answers:
A) 29616 pounds  
B) $$ f_{x} = 14500 + 10xy - 21x^2 $$  
C) $$ f_{y} = 3000 + 5x^2 $$  
D) 15111 weekly pounds per skilled worker  
E) 3845 weekly pounds per unskilled worker
A) 29616 pounds B) $$ f_{x} = 14500 + 10xy - 21x^2 $$ C) $$ f_{y} = 3000 + 5x^2 $$ D) 15111 weekly pounds per skilled worker E) 3845 weekly pounds per unskilled worker

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