Problem 7 - Basic Technical Mathematics with Calculus

Question

The velocity (in \(\mathrm{km} / \mathrm{h}\) ) of a plane flying into an increasing headwind is \(v=50(12-t),\) where \(t\) is the time (in h). How far does the plane travel in a 2.0 -h trip?

Step-by-Step Solution

Answer

The plane travels 1100 kilometers in a 2-hour trip.

Step 1: Identifying the Given Function

The velocity of the plane is given by the function \( v(t) = 50(12-t) \), where \( t \) is time in hours.

Step 2: Determine the Limits of Integration

The problem states that the plane travels for 2 hours, meaning our limits of integration will be from \( t = 0 \) to \( t = 2 \). This will allow us to find the total distance traveled over this time period.

Step 3: Set up the Integral for Distance Calculation

Distance traveled is the integral of velocity over time. Set up the integral: \[ \int_{0}^{2} 50(12-t)\, dt \]

Step 4: Integrate the Function

Calculate the integral \( \int 50(12-t) \, dt \):1. Find the antiderivative: \( 50 \int (12-t) \, dt = 50 \left( 12t - \frac{t^2}{2} \right) \).2. Evaluate it from \( t=0 \) to \( t=2 \).

Step 5: Evaluate the Definite Integral

Substitute the limits into the antiderivative:\( 50 \left[ (12 \times 2 - \frac{2^2}{2}) - (12 \times 0 - \frac{0^2}{2}) \right] = 50 \left[ (24 - 2) - 0 \right] = 50 \times 22 \).

Step 6: Calculate the Final Distance

Multiply the final expression to find the total distance: \( 50 \times 22 = 1100 \) kilometers.

Key Concepts

Definite Integral

The definite integral is a fundamental concept in calculus used to calculate the accumulation of quantities. In the context of this problem, it allows us to determine the total distance the plane travels over a specified time period. To achieve this, we set up an integral, which represents the sum of continuously added quantities—in this case, the velocity over time.

• The integral symbol, \( \int \), represents the operation of summing velocities.
• The limits of integration, from \( t = 0 \) to \( t = 2 \), indicate that we are interested in the velocity from the start of the journey to 2 hours later.
• At the core, integration is the reverse operation of differentiation and helps us understand the accumulated effect of changing quantities.

In practice, solving the integral gives you the total change—here, how far the plane travels in the 2-hour period.

Velocity Function

A velocity function describes how the velocity of an object changes over time. For the plane in this exercise, the velocity is given by the function \( v(t) = 50(12-t) \). This equation shows that the velocity depends linearly on time.

• The coefficient 50 indicates that the base rate of change is 50 km/h for each unit of the expression \( (12-t) \).
• This tells us that as time \( t \) increases, the expression \( (12-t) \) decreases, meaning the plane slows down—due to the increasing headwind.

Understanding the velocity function is essential because it directly influences the calculation of distance. The changes in velocity translate into variations in the distance covered over time.

Distance Calculation

Distance is calculated as the integral of the velocity function over the time interval of interest. In this exercise, we use the given velocity function to find the exact distance the plane travels.

• Set up the integral stated earlier: \( \int_{0}^{2} 50(12-t) \, dt \). This represents the total distance traveled from \( t = 0 \) to \( t = 2 \) hours.
• To compute the integral, find the antiderivative of the velocity function, which is the reverse process of differentiation.
• The antiderivative is evaluated between the limits of 0 and 2. For our velocity function, this results in the expression \( 50 \left( 12t - \frac{t^2}{2} \right) \).

Substituting the limits and simplifying, we compute the total distance as 1100 kilometers. This integral calculation assures that every minute and second of the 2-hour journey is accounted for in the total distance traveled.

Limits of Integration

The limits of integration determine the scope or range over which we compute an integral. They specify the starting and ending points for the integration process. For the plane's journey, these limits are critical in calculating the exact path.

• In this problem, the limits are \( t = 0 \) and \( t = 2 \), marking the start and end of the 2-hour flight.
• These limits ensure that we calculate the total accumulated effect of velocity from the beginning to the end of the specified time.
• Employing proper limits ensures the definite integral accurately reflects the total distance or change over the interval.

Using these precise limits on our velocity function integral provides a complete picture of the journey, covering every moment within the defined period.