Problem 73 - Precalculus with Limits

Question

REVENUE The vector \(\mathbf{u} = \langle 4600, 5260 \rangle\) gives the numbers of units of two models of cellular phones produced by a telecommunications company. The vector \(\mathbf{v} = \langle 79.99, 99.99 \rangle\) gives the prices (in dollars) of the two models of cellular phones, respectively. (a) Find the dot product \(\mathbf{u} \cdot \mathbf{v}\) and interpret the result in the context of the problem. (b) Identify the vector operation used to increase the prices by 5%.

Step-by-Step Solution

Answer

The dot product \(\mathbf{u} \cdot \mathbf{v}\) is 891297.40. This represents the total revenue from selling the units of cellular phones. To increase the prices of the units by 5%, the vector operation used is scalar multiplication, resulting in a new price vector \(\langle 83.99, 104.99 \rangle\).

Step 1: Compute the dot product

The dot product of two vectors is computed as the sum of the products of their corresponding components. In this case, \(\mathbf{u} \cdot \mathbf{v} = u_1*v_1 + u_2*v_2 = (4600*79.99) + (5260*99.99) = 365354.00 + 525943.40 = 891297.40.\) This value represents the total revenue from selling the units of cellular phones.

Step 2: Increase the prices by 5%

To increase the prices by 5%, we use scalar multiplication. We multiply the price vector \(\mathbf{v}\) by 1.05 (representing a 5% increase). This gives a new vector: \(1.05*\mathbf{v} = \langle 1.05*79.99, 1.05*99.99 \rangle = \langle 83.99, 104.99 \rangle\). Each component of the new vector represents the increased price for the respective model of cellular phone.

Key Concepts

Dot Product

The concept of the dot product is a fundamental operation in vector mathematics. To calculate the dot product of two vectors, you multiply their corresponding components and sum the results. This operation is useful for determining various financial and geometric calculations.

Given the vectors \( \mathbf{u} = \langle 4600, 5260 \rangle \) and \( \mathbf{v} = \langle 79.99, 99.99 \rangle \), we use the formula:

• \( \mathbf{u} \cdot \mathbf{v} = u_1 \times v_1 + u_2 \times v_2 \)

Substituting the values, the calculation becomes:

• \( 4600 \times 79.99 + 5260 \times 99.99 = 365354.00 + 525943.40 = 891297.40 \)

The result, \( 891297.40 \), represents the total revenue generated from the sale of both types of cellular phones. This is because each pair of numbers in vector \( \mathbf{u} \) (units produced) and vector \( \mathbf{v} \) (price per unit) corresponds to a specific model, and their product is the revenue from that model. Therefore, adding them gives the company’s total revenue from those sales.

Scalar Multiplication

Scalar multiplication is a simple yet powerful vector operation that scales a vector by multiplying each of its components by a constant, known as a scalar.

In the context of this problem, we're using scalar multiplication to model an increase in prices by 5%. The current prices are represented by \( \mathbf{v} = \langle 79.99, 99.99 \rangle \). To increase these prices by 5%, you apply the scalar \( 1.05 \):

• Multiply each component of \( \mathbf{v} \) by \( 1.05 \).
• \( 1.05 \times 79.99 = 83.99 \)
• \( 1.05 \times 99.99 = 104.99 \)

This yields a new vector, \( \langle 83.99, 104.99 \rangle \), where each component reflects the increased prices of the cellular phone models.

Scalar multiplication is widely used in economics and finance to model scenarios like price adjustments, inflation impacts, and scaling projections. By understanding this operation, you can easily adapt models to account for changing factors, such as percentage increases or decreases in different contexts.

Total Revenue Calculation

Total revenue calculation involves integrating different factors to understand the financial outcome. In scenarios like this one, vectors are used to succinctly combine quantities produced with their corresponding prices.

The procedure follows these general steps:

• Identify the vector that represents units of goods produced \( \mathbf{u} \).
• Identify the vector that represents the selling prices \( \mathbf{v} \).
• Use the dot product to calculate total revenue: \( \mathbf{u} \cdot \mathbf{v} \).
• The sum of products gives the total revenue from all units sold.

Understanding and applying this methodology is crucial in business analysis, as it aggregates complex data into usable insights efficiently.

For any business producing multiple products, recognizing and implementing vector operations can facilitate quick calculations and strategic decision-making processes. This approach allows managers to project revenues under different sales scenarios and pricing strategies, aiding in comprehensive financial planning.