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Chapter 6: Problem 51

The components of \(\mathbf{v}=240 \mathbf{i}+300 \mathbf{j}\) represent the respective number of gallons of regular and premium gas sold at a station. The components of \(\mathbf{w}=2.90 \mathbf{i}+3.07 \mathbf{j}\) represent the respective prices per gallon for each kind of gas. Find \(\mathbf{v} \cdot \mathbf{w}\) and describe what the answer means in practical terms.

Short Answer

Expert verified
The dot product \(\mathbf{v} \cdot \mathbf{w} = 1617.0\). This implies that the total sales at the gas station amount to $1617.

Step by step solution

01

Define the Vectors

The vector \(\mathbf{v}\) is the quantities of regular and premium gas sold, given as \(240\mathbf{i} + 300\mathbf{j}\). The vector \(\mathbf{w}\) is the price per gallon of each type of gas, represented as \(2.90\mathbf{i} + 3.07\mathbf{j}\).
02

Compute the Dot Product

To compute the dot product \(\mathbf{v} \cdot \mathbf{w}\), multiply corresponding components together and sum the results. Therefore, \(\mathbf{v} \cdot \mathbf{w} = (240 \times 2.90) + (300 \times 3.07)\).
03

Calculate the Total

Evaluating the expression gives us the total value of gas sold. Thus, \(\mathbf{v} \cdot \mathbf{w} = 696.0 + 921.0 = 1617.0\)

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