What is the quotient of the complex number 4-3i divided by its conjugate

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A. What is the quotient of the complex number 4-3i divided by its conjugate?

B. expected to vary between 19 and 30 couples each month. Clot the icon boo view the rewrindor sharpie spradihet The demand must the at west cogs for the gasoline mirth-mart ion breath wave. (f)one a white number) Newsvendor model spreadsheet The formula for the quantity sold is =MIN(B11,B12). The formula for the surplus quantity is =MAX(0,B12-B11). The formula for the profit is =B14* B5-B12* B6.

The correct answer and explanation is:

Part A: Quotient of a Complex Number Divided by its Conjugate

The given complex number is z=4−3iz = 4 – 3i, and its conjugate is z‾=4+3i\overline{z} = 4 + 3i.

The formula to divide a complex number by its conjugate is: zz‾=4−3i4+3i.\frac{z}{\overline{z}} = \frac{4 – 3i}{4 + 3i}.

To perform the division, multiply the numerator and denominator by the conjugate of the denominator: 4−3i4+3i⋅4−3i4−3i=(4−3i)2(4)2+(3)2.\frac{4 – 3i}{4 + 3i} \cdot \frac{4 – 3i}{4 – 3i} = \frac{(4 – 3i)^2}{(4)^2 + (3)^2}.

  1. Simplify the numerator: (4−3i)2=16−24i+9i2=16−24i−9=7−24i.(4 – 3i)^2 = 16 – 24i + 9i^2 = 16 – 24i – 9 = 7 – 24i.
  2. Simplify the denominator: (4)2+(3)2=16+9=25.(4)^2 + (3)^2 = 16 + 9 = 25.
  3. Combine: 4−3i4+3i=7−24i25=725−24i25.\frac{4 – 3i}{4 + 3i} = \frac{7 – 24i}{25} = \frac{7}{25} – \frac{24i}{25}.

The quotient is: 725−24i25.\boxed{\frac{7}{25} – \frac{24i}{25}}.

Part B: Newsvendor Model Explanation

The Newsvendor Model helps in deciding optimal inventory levels to maximize profits when demand is uncertain. Here’s how the spreadsheet formulas relate to the scenario:

  1. Demand and Quantity Sold: The demand varies between 19 and 30 couples monthly.
    The formula for the quantity sold is: Quantity Sold=MIN(B11, B12),\text{Quantity Sold} = \text{MIN(B11, B12)}, where:
    • B11B11 represents the demand,
    • B12B12 represents the quantity ordered.
    This ensures the quantity sold equals the smaller of the demand or the ordered quantity.
  2. Surplus Quantity: Surplus inventory occurs if the ordered quantity exceeds demand.
    The formula for surplus is: Surplus Quantity=MAX(0,B12−B11).\text{Surplus Quantity} = \text{MAX}(0, B12 – B11).
  3. Profit Calculation: The profit formula is: Profit=(B14×B5)−(B12×B6),\text{Profit} = (B14 \times B5) – (B12 \times B6), where:
    • B14B14: Quantity sold,
    • B5B5: Selling price per unit,
    • B12B12: Ordered quantity,
    • B6B6: Cost per unit.
  4. Optimization Insight: The profit depends on balancing the cost of surplus and missed sales opportunities.
    • Ordering too much incurs surplus costs (storage or waste).
    • Ordering too little results in missed sales.

By using these formulas, businesses can determine the optimal order quantity under uncertain demand, ensuring minimal waste while maximizing profits.

Summary

The Newsvendor Model aligns inventory with demand, optimizing profits and reducing surplus or missed sales. Understanding and applying the formulas ensure better inventory decisions in dynamic market conditions.

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