Problem 74
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
Verified Answer
The dot product \(\mathbf{u} \cdot \mathbf{v}\) is 10137.5 dollars. This is the total revenue of selling all the hamburgers and hot dogs at their respective prices. The operation to increase prices by 2.5% is scalar multiplication.
1Step 1: Finding the Dot Product
To find the dot product \(\mathbf{u} \cdot \mathbf{v}\), simply carry out the operation: \(3140 \times 2.25 + 2750 \times 1.75\).
2Step 2: Interpreting the Result
The result of the dot product operation from step 1 represents the total revenue generated from the sale of hamburgers and hot dogs in one month. This assumes that all the food was sold at their respective prices
3Step 3: Increasing the Prices
The vector operation that would be used to increase the prices by 2.5% is scalar multiplication. \(\mathbf{v}_{new} = 1.025 \times \mathbf{v}\), this means each component of the price vector \(\mathbf{v}\) will be multiplied by 1.025, which is equivalent to a 2.5% increase.
Key Concepts
Vector OperationsScalar MultiplicationTotal Revenue Calculation
Vector Operations
Vector operations are fundamental to understanding many mathematical and real-world applications. The dot product is a crucial type of vector operation which combines two vectors and results in a single number, known as a scalar. This operation is performed by multiplying corresponding components of two vectors and then summing the results. In our exercise, the dot product of the vectors \( \mathbf{u} \) and \( \mathbf{v} \) is found by multiplying the numbers of hamburgers and hot dogs sold by their respective prices.
For example, if \( \mathbf{u} = \langle x_1, y_1 \rangle \) and \( \mathbf{v} = \langle x_2, y_2 \rangle \) are two vectors, the dot product is calculated as \( x_1 \times x_2 + y_1 \times y_2 \).
This operation is not only used in business for revenue calculations but also in physics for calculating work done and in computer science for graphics and machine learning algorithms.
For example, if \( \mathbf{u} = \langle x_1, y_1 \rangle \) and \( \mathbf{v} = \langle x_2, y_2 \rangle \) are two vectors, the dot product is calculated as \( x_1 \times x_2 + y_1 \times y_2 \).
This operation is not only used in business for revenue calculations but also in physics for calculating work done and in computer science for graphics and machine learning algorithms.
Scalar Multiplication
Scalar multiplication involves multiplying a vector by a single number (a scalar), and it uniformly scales the vector's magnitudes while keeping the direction unchanged if the scalar is positive. When we refer to 'increasing prices by 2.5%', we implicitly talk about applying scalar multiplication to the price vector. In this context, each price element in vector \( \mathbf{v} \) is multiplied by 1.025, representing a 2.5% increase.
For instance, if we have a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \) and a scalar \( k \), the scalar multiplication is performed as \( k \times \mathbf{a} = \langle k \times a_1, k \times a_2 \rangle \).
This operation is important in various areas such as physics to represent forces and velocities, economics to adjust for inflation, and in vector graphics where it's used to scale images.
For instance, if we have a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \) and a scalar \( k \), the scalar multiplication is performed as \( k \times \mathbf{a} = \langle k \times a_1, k \times a_2 \rangle \).
This operation is important in various areas such as physics to represent forces and velocities, economics to adjust for inflation, and in vector graphics where it's used to scale images.
Total Revenue Calculation
In a business context, the total revenue is the total amount of money made from selling products or services. It is calculated by multiplying the quantity sold with the price of the individual items.
Let's take our exercise as an example. The dot product \( \mathbf{u} \cdot \mathbf{v} \) essentially calculated the total revenue by performing this product on a component-wise level. This meant for every hamburger sold at a price of \(2.25 and every hot dog sold at a price of \)1.75, the total revenue is the sum of these individual sales.
Let's take our exercise as an example. The dot product \( \mathbf{u} \cdot \mathbf{v} \) essentially calculated the total revenue by performing this product on a component-wise level. This meant for every hamburger sold at a price of \(2.25 and every hot dog sold at a price of \)1.75, the total revenue is the sum of these individual sales.
- Total revenue from hamburgers: \( 3140 \times 2.25 \) dollars
- Total revenue from hot dogs: \( 2750 \times 1.75 \) dollars
- Total combined revenue: The sum of the above
Other exercises in this chapter
Problem 73
Business The vector \(\mathbf{u}=\langle 1225,2445\rangle\) gives the numbers of hours worked by employees of a temp agency at two pay levels. The vector \(\mat
View solution Problem 73
Finding a Power of a Complex Number In Exercises \(65-80\) , use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in stand
View solution Problem 74
Finding the Component Form of a Vector In Exercises \(67-74\) , find the component form of \(v\) given its magnitude and the angle it makes with the positive \(
View solution Problem 74
Finding a Power of a Complex Number In Exercises \(65-80\) , use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in stand
View solution