Problem 74 - Precalculus with Limits

Question

REVENUE The vector \(\mathbf{u} = \langle 3140, 2750 \rangle\) gives the numbers of hamburgers and hot dogs, respectively, sold at a fast-food stand in one month. The vector \(\mathbf{v} = \langle 2.25, 1.75 \rangle\) gives the prices (in dollars) of the food items. (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 2.5%.

Step-by-Step Solution

Answer

The dot product \(\mathbf{u} \cdot \mathbf{v}\) gives the total revenue for the month while multiplying the vector \(\mathbf{v}\) with the scalar 1.025 to reflect the increase in prices by 2.5%.

Step 1: Compute the Dot Product

The given vectors are \(\mathbf{u} = \langle 3140, 2750 \rangle\) and \(\mathbf{v} = \langle 2.25, 1.75 \rangle\). Typically, the dot product of two vectors \(\mathbf{u} = \langle u_1, u_2 \rangle\) and \(\mathbf{v} = \langle v_1, v_2 \rangle\) is computed using the formula \(\mathbf{u} \cdot \mathbf{v} = u_1*v_1 + u_2*v_2\). In this case, \(\mathbf{u} \cdot \mathbf{v} = 3140*2.25 + 2750*1.75\).

Step 2: Interpret the Dot Product

The result in the context of the problem gives the total revenue earned from selling hamburgers and hot dogs in one month. The calculation from step 1 gives the revenue in dollars.

Step 3: Identify the Vector Operation

To increase the prices by 2.5%, we can use the operation of scalar multiplication on vector \(\mathbf{v}\). The operation is represented as \(1.025*\mathbf{v} = \langle 1.025*2.25, 1.025*1.75 \rangle\).

Key Concepts

Dot Product

In vector mathematics, the dot product is a fundamental operation that calculates the sum of the products of the corresponding components of two vectors. It's often used to find things like projections and to work with angles between vectors.
When we take vectors \( \mathbf{u} = \langle 3140, 2750 \rangle \) and \( \mathbf{v} = \langle 2.25, 1.75 \rangle \), each element in these vectors has a specific context. For \( \mathbf{u} \), the elements represent quantities of hamburgers and hot dogs sold. For \( \mathbf{v} \), they are prices in dollars.

The formula for the dot product is:
\( \mathbf{u} \cdot \mathbf{v} = u_1 \cdot v_1 + u_2 \cdot v_2 \).
So, calculating this gives:
\[ \mathbf{u} \cdot \mathbf{v} = 3140 \cdot 2.25 + 2750 \cdot 1.75 \]

• 3140 hamburgers sold at \(2.25 each.
• 2750 hot dogs sold at \)1.75 each.

This results in a total revenue, showing how the product of quantities and prices gives the earnings from these sales.

Scalar Multiplication

Scalar multiplication in vector mathematics is an operation where each component of a vector is multiplied by the same scalar value. This process is helpful for scaling vectors, adjusting their magnitude while maintaining their direction.
To adjust the prices in our original vector \( \mathbf{v} = \langle 2.25, 1.75 \rangle \) by 2.5%, scalar multiplication is used. The scalar value in this context is 1.025 (or 102.5% of the original price).

The operation to increase the prices would look like:
\( 1.025 \times \mathbf{v} = \langle 1.025 \times 2.25, 1.025 \times 1.75 \rangle \)
This converts to adjusted prices:

• New hamburger price: \( 1.025 \times 2.25 = 2.30625 \)
• New hot dog price: \( 1.025 \times 1.75 = 1.79375 \)

Using scalar multiplication, we apply a consistent percentage increase, making it a practical tool for price adjustments in economics.

Revenue Calculation

Calculating revenue using vectors simplifies the process of determining total earnings from multiple products by utilizing mathematical operations on vectors.
The dot product, as demonstrated, is employed to compute revenue. In this context, it combines sales quantities and individual item prices to produce a single figure representing total revenue.

The formula derived: \[ \mathbf{u} \cdot \mathbf{v} = 3140 \times 2.25 + 2750 \times 1.75 \] gives us the comprehensive total in dollars.
In business contexts, it helps in:

• Summing earnings from multiple items quickly.
• Integration into financial software for instant calculations.

Revenue calculations powered by vectors thus streamline financial assessment, providing clear, concise data for decision-making.