Question

After being in business for \(t\) years, a manufacturer of cars is producing \(120+2 t+3 t^{2}\) units per year. The sales price in dollars per unit has risen according to the formula \(6000+700 t\). Write a formula for the manufacturer's yearly revenue \(R(t)\) after \(t\) years.

Short answer

The revenue formula is \(R(t) = 720,000 + 96,000t + 19,400t^2 + 2,100t^3\).

Step-by-step solution

Identify the Components of Revenue

The revenue of a business is calculated by multiplying the number of units sold by the price per unit. In this problem, the number of units produced per year after \(t\) years is given by \(120 + 2t + 3t^2\), and the price per unit is \(6000 + 700t\).

Set Up the Revenue Formula

The formula for revenue \(R(t)\) is determined by multiplying the number of units produced by the price per unit. Thus, \(R(t) = (120 + 2t + 3t^2) \times (6000 + 700t)\).

Expand the Revenue Formula

Expand the product \((120 + 2t + 3t^2)(6000 + 700t)\) using the distributive property (also known as FOIL for binomials):- First, multiply \(120\) by \(6000 + 700t\).- Then, multiply \(2t\) by \(6000 + 700t\).- Finally, multiply \(3t^2\) by \(6000 + 700t\).This results in:\[R(t) = 120 \times 6000 + 120 \times 700t + 2t \times 6000 + 2t \times 700t + 3t^2 \times 6000 + 3t^2 \times 700t\].

Simplify Each Multiplication

Calculate each component of the expanded formula:- \(120 \times 6000 = 720,000\)- \(120 \times 700t = 84,000t\)- \(2t \times 6000 = 12,000t\)- \(2t \times 700t = 1,400t^2\)- \(3t^2 \times 6000 = 18,000t^2\)- \(3t^2 \times 700t = 2,100t^3\)Add these together to form:\[R(t) = 720,000 + 84,000t + 12,000t + 1,400t^2 + 18,000t^2 + 2,100t^3\].

Combine Like Terms

Combine the terms with similar powers of \(t\):- The constant term: \(720,000\)- The \(t\) terms: \(84,000t + 12,000t = 96,000t\)- The \(t^2\) terms: \(1,400t^2 + 18,000t^2 = 19,400t^2\)- The \(t^3\) term: \(2,100t^3\)Thus, the revenue function is:\[R(t) = 720,000 + 96,000t + 19,400t^2 + 2,100t^3\].

Key concepts

Revenue function

Understanding the concept of a revenue function is essential in economics and business. It tells us how much money a company earns from selling its products. In our exercise, the revenue function is about determining the annual income a car manufacturer earns based on production and price.

The revenue function is calculated by multiplying two main factors:

• The number of units sold each year, which changes over time as given by: \(120 + 2t + 3t^2\).
• The selling price per unit also changes over time as given by: \(6000 + 700t\).

So, the formula for the revenue \(R(t)\) after \(t\) years is obtained by multiplying these two expressions:\[R(t) = (120 + 2t + 3t^2)(6000 + 700t)\].This gives an expression that reflects how revenue evolves as time passes.

Polynomial expansion

Polynomial expansion is a mathematical process used to express a product of polynomials as a sum of terms. It involves spreading out terms in a mathematical expression to make them more manageable and easy to simplify.

When we expand polynomials, like in our revenue function example, we are looking at two large expressions multiplied together:\((120 + 2t + 3t^2)\) and \((6000 + 700t)\).

Breaking it down makes it simpler to compute. The expanded form is achieved by multiplying every term in the first polynomial by every term in the second polynomial. This process helps us see how each term contributes to the final revenue calculation, producing a detailed picture of potential earnings over time.

Distributive property

The distributive property is a foundational algebraic principle. It states that multiplying a number by a sum is the same as multiplying each addend separately and then adding the products. For example, multiplying \(a\) by \((b + c)\) gives us \[a \times (b + c) = ab + ac\].

In our exercise, we apply this property to expand the revenue function \((120 + 2t + 3t^2)(6000 + 700t)\).
Here's how it works in steps:

• Multiply \(120\) by each term in \((6000 + 700t)\).
• Repeat the process for \(2t\) and for \(3t^2\).

This careful, step-by-step multiplication helps to arrive at the complete expanded expression of the revenue function, which is crucial for understanding the company's financial prospects over time.

Algebraic expressions

Algebraic expressions consist of numbers, variables, and operations. They are crucial in representing real-world scenarios mathematically, such as calculating a company's revenue.

In this scenario, the algebraic expressions represent production and pricing:

• Production: \(120 + 2t + 3t^2\)
• Price: \(6000 + 700t\)

Algebraic expressions allow us to model and solve problems by rearranging and simplifying terms. In this problem, simplifying these expressions and multiplying them gives the revenue model. Understanding these expressions helps one see how various factors interplay to influence the overall revenue outcome.