Evaluating Formulas and Forecasting Vehicle Sales

Evaluating Algebraic Formulas

• Module Reference: Mod E1; sec 4.3 notes.
• General Equation: $M = P(1 + RT)$
  • $M$: Mature value
  • $P$: Principal amount
  • $R$: Rate
  • $T$: Time
• Given Values:
  • $M = \$12,000$
  • $T = 0.5$
• Objective: Solve for the rate ($R$) given $M = \$12,000$ and $T = 0.5$ in the equation $M = P(1 + RT)$.
• Steps to Solve:
  • Use the distributive property on the equation.
  • Subtract $12,000$ from each side.
• Solution:
  • Rate ($R$) in decimal form: $0.045$
  • Rate ($R$) in percentage form: $4.5\%$

Solving Literal Equations for Specific Variables

• Formula Overview: This formula determines the discount ($D$) of a note based on its related parameters:
  • $B$: Face value of the note
  • $T$: Time of the note
  • $M$: Mature value
• Objective: Solve for the time variable ($T$) in the formula.
• Step-by-Step Isolation Procedure:
  • Multiply both sides of the equation by $MT$
  • Divide both sides of the resulting equation by $DM$

Case Study: Forecasting Monthly Vehicle Sales

• Context: Ben James believes that the dollar amount of Chevrolet vehicles sold in one state can be approximated using a mathematical forecasting equation.
• Unit Specifications:
  • Advertising expenditures are expressed in millions of dollars.
  • Vehicle sales figures are expressed in millions of dollars.
• July Sales Forecasting Scenario:
  • Month evaluated: July
  • Advertising input parameter: $\$1.4\,\text{million}$
• Evaluation Process:
  • Step 1: Evaluate the exponent term first.
  • Step 2: Multiply the result.
• Model Forecast Result:
  • Based on the model, projected sales for July are forecasted to be $\$71,752,000$