Problem 12 - Technical Mathematics with Calculus

Question

A certain automobile engine delivers 53 hp and has a displacement (the total volume swept out by the pistons) of 3.0 liters. If the power is directly proportional to the displacement, what horsepower would you expect from a similar engine that has a displacement of 3.8 liters?

Step-by-Step Solution

Answer

The expected horsepower for the second engine with a displacement of 3.8 liters is approximately 67.27 hp.

Step 1: Understand the Relationship Between Power and Displacement

Given that power (P) is directly proportional to displacement (V), we can express this relationship as P = kV, where k is the constant of proportionality.

Step 2: Calculate the Constant of Proportionality (k)

To find the constant k, use the information given for the first engine. Substitute the known values of P (53 hp) and V (3.0 liters) into the equation to get 53 hp = k * 3.0 liters. Solve for k by dividing both sides by 3.0 liters to get k = 53 hp / 3.0 liters.

Step 3: Apply the Constant of Proportionality to the Second Engine

Using the value calculated for k, predict the horsepower for an engine with a displacement of 3.8 liters. P = k * V2, where V2 is the displacement of the second engine. Substitute the known values into the equation to find P.

Step 4: Calculate the Expected Horsepower for the Second Engine

Substitute k and the displacement of the second engine (3.8 liters) into the equation from step 3 to calculate the expected horsepower: P = (53 hp / 3.0 liters) * 3.8 liters.

Key Concepts

Direct Proportionality

In physics, direct proportionality is a fundamental concept that describes a relationship between two variables where they increase or decrease at the same rate. The principle is quite straightforward: when one variable doubles, the other doubles as well; if one halves, so does the other.

Take the example of an automobile engine where its power is said to be directly proportional to the engine's displacement. This means that as the displacement of the engine increases, the engine's power outputs also increase, maintaining a constant ratio between the two. To visualize it, imagine a graph with displacement on the x-axis and power on the y-axis: a straight line through the origin demonstrates this direct relationship.

One real-life implication of direct proportionality is in designing engines since it allows engineers to predict how changes in engine size can affect power output. This is crucial for achieving desired performance levels in various types of vehicles.

Constant of Proportionality

The constant of proportionality, denoted as 'k' in equations, is the factor that connects two variables that are directly proportional to each other. It represents the ratio of the two quantities that remains the same as they are varied.

In the context of the engine power and displacement relationship, using the formula 'P = kV', where 'P' represents power, 'V' the displacement, and 'k' the constant of proportionality, it becomes possible to compare different engines. By determining the constant of proportionality from a known engine's power and displacement, you can predict the power of a similar engine with a different displacement.

For example, once 'k' is found using the formula with a known engine's data, that same 'k' can be used for predicting the power of any other similar engine, just by knowing its displacement. The constant of proportionality, thus, acts like a bridge, enabling these predictions and comparisons.

Power to Displacement Relationship

Exploring the power to displacement relationship in engines showcases the practical application of direct proportionality. Power, generally measured in horsepower (hp) for engines, describes the rate at which work is done, while displacement refers to the volume swept by the pistons within the engine cylinders, measured in liters or cubic centimeters.

This relationship is crucial for engine manufacturers and automotive designers, as it guides the design of engines with the desired power output. In our exercise, the relationship helps us deduce that an engine with a larger displacement is expected to produce more power. Consequently, knowing the power output of one engine and its displacement lets us calculate the expected power output of another engine with a different displacement by using the constant of proportionality.

Through such calculations, we can optimize engines for specific tasks, like providing more power for a sports car or achieving fuel efficiency in a family sedan, by balancing the power needed with the appropriate engine displacement.