Problem 81 - Precalculus

Question

Use the following tables. The following table gives fuel and electric requirements per mile associated with gasoline and electric automobiles: $$\begin{array}{|l|c|c|}\hline & \begin{array}{c} \text { Numeer of } \\\\\text { GatLons/Mile }\end{array} & \begin{array}{c}\text { Numeer of } \\\\\text { kW-hr/Mile }\end{array} \\\\\hline \text { SUV full size } & 0.06 & 0 \\ \hline \text { Hybrid car } & 0.02 & 0.1 \\\\\hline \text { Electric car } & 0 & 0.3 \\\\\hline\end{array}$$ The following table gives an average cost for gasoline and electricity: $$\begin{array}{|l|c|}\hline \text { Cost per gallon of gasoline } & \$ 3.80 \\\\\hline \text { Cost per kW-hr of electricity } & \$ 0.05 \\\\\hline\end{array}$$ Environment. Let matrix \(A\) represent the gasoline and electricity consumption and matrix \(B\) represent the costs of gasoline and electricity. Find \(A B\) and describe what the elements of the product matrix represent. (Hint: \(A\) has order \(3 \times 2\) and \(B\) has order \(2 \times 1 .)\)

Step-by-Step Solution

Answer

\(AB = \begin{pmatrix} 0.228 \\ 0.076 \\ 0.015 \end{pmatrix}\); each element represents the cost per mile for each vehicle type.

Step 1: Identify matrices A and B

Matrix \(A\), representing consumption, is given by the following 3x2 matrix: \[A = \begin{pmatrix} 0.06 & 0 \0.02 & 0.1 \0 & 0.3 \\end{pmatrix} \]Matrix \(B\), representing cost, is given by the following 2x1 matrix: \[B = \begin{pmatrix} 3.80 \0.05 \end{pmatrix} \]

Step 2: Multiply Matrices A and B

To find \(AB\), multiply each element of row from \(A\) with the corresponding element of column from \(B\), then sum these products for each entry.**Calculate for the first row of the product matrix:**\(0.06 \times 3.80 + 0 \times 0.05 = 0.228\)**Calculate for the second row of the product matrix:**\(0.02 \times 3.80 + 0.1 \times 0.05 = 0.076\)**Calculate for the third row of the product matrix:**\(0 \times 3.80 + 0.3 \times 0.05 = 0.015\)

Step 3: Write the Resulting Matrix

The resulting product matrix \(AB\) is:\[AB = \begin{pmatrix} 0.228 \0.076 \0.015 \end{pmatrix} \]

Step 4: Interpret the Elements of the Product Matrix

Each element of the product matrix \(AB\) represents the cost per mile of operating each type of vehicle:- \(0.228\) represents the cost per mile for the SUV.- \(0.076\) represents the cost per mile for the Hybrid car.- \(0.015\) represents the cost per mile for the Electric car.

Key Concepts

Fuel Consumption in Vehicles

Fuel consumption refers to the amount of fuel a vehicle uses to travel a certain distance. It's a crucial factor when calculating the operating cost and environmental impact of different vehicles. In this context, fuel consumption can include both traditional gasoline and electricity for electric vehicles. Each vehicle type, such as SUV, hybrid, and electric cars, has distinct consumption rates.

• SUVs typically consume more fuel, needing around 0.06 gallons of gasoline per mile.
• Hybrid cars use both gasoline and electricity, requiring 0.02 gallons and 0.1 kW-hr per mile, respectively.
• All-electric vehicles rely solely on electricity, consuming about 0.3 kW-hr per mile.

Understanding these figures helps car owners and potential buyers evaluate the efficiency and cost-effectiveness of different vehicles.

Cost Calculation Based on Fuel Type

Cost calculation involves determining the expense associated with using fuel or electricity. This is typically expressed as a cost per mile to conveniently compare across vehicle types. To calculate this, we use fuel consumption values and multiply them by the cost per unit of fuel or electricity.

• For gasoline, the average cost is $3.80 per gallon.
• For electricity, it's $0.05 per kW-hr.

By multiplying these costs with the respective consumption rates for each vehicle type, we can derive the cost per mile. For instance, SUVs incur a higher cost per mile due to their higher fuel consumption, while electric vehicles benefit from lower electricity prices, resulting in minimal cost per mile.

Advantages of Electric Vehicles

Electric vehicles (EVs) represent a significant shift in how we think about transportation. They are designed to run solely on electricity, which offers numerous benefits. First and foremost, they typically have much lower operating costs per mile compared to gasoline vehicles.

• EVs have zero gasoline consumption, relying entirely on electricity.
• With costs as low as $0.05 per kW-hr of electricity, their per-mile cost can be as low as $0.015.
• This economy encourages both cost savings for owners and reduced environmental impact due to lower emissions.

Additionally, EVs help reduce the dependence on fossil fuels, offering a cleaner, sustainable alternative tailored to modern ecological goals.