Problem 51 - Algebra and Trigonometry

Question

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.

Step-by-Step Solution

Answer

The result of the dot product \(\mathbf{v} \cdot \mathbf{w}\) is 1617, which represents the total income in dollars from the sale of both kinds of gas.

Step 1: Identify the Components of Each Vector

The components of each vector are given by: \(\mathbf{v}=240 \mathbf{i}+300 \mathbf{j}\) and \(\mathbf{w}=2.90 \mathbf{i}+3.07 \mathbf{j}\). That means there are 240 gallons of regular gas sold at $2.90 per gallon (i-component of the vector) and 300 gallons of premium gas sold at $3.07 per gallon (j-component of the vector).

Step 2: Compute the Dot Product Between \(\mathbf{v}\) and \(\mathbf{w}\)

The dot product of two vectors can be obtained by multiplying corresponding components and adding the result. This gives \(\mathbf{v} \cdot \mathbf{w} = (240)(2.90) + (300)(3.07)\)

Step 3: Do the Calculations and Obtain the Result

By performing the calculations we get \(\mathbf{v} \cdot \mathbf{w} = 696 + 921 = 1617\).