Problem 18 - Calculus Early Transcendentals

Question

On the first day of the year \((t=0),\) a city uses electricity at a rate of \(2000 \mathrm{MW}\). That rate is projected to increase at a rate of \(1.3 \%\) per year. a. Based on these figures, find an exponential growth function for the power (rate of electricity use) for the city. b. Find the total energy (in \(\mathrm{MW}\) -yr) used by the city over four full years beginning at \(t=0\). c. Find a function that gives the total energy used (in \(\mathrm{MW}\) -yr) between \(t=0\) and any future time \(t > 0\).

Step-by-Step Solution

Answer

Answer: The exponential function representing the city's power consumption rate is P(t) = 2000 * (1 + 0.013)^t. The total energy used by the city over four years is approximately 8198.36 MW-yr. The function E(t) that gives the total energy used between t=0 and any future time t>0 is E(t) ≈ 153846.15 * ((1.013^(t+1))/(t+1)) - 153846.15.

Step 1: Finding the exponential growth function for the power rate

The initial power rate is given as 2000 MW when t=0, and the rate is growing at 1.3% per year. We need to find an exponential growth function for the power rate, P(t), for the city. In general, the exponential growth function can be written as: P(t) = P_0 * (1 + r)^t Where P_0 is the initial power rate, r is the percentage growth rate and t represents the time in years. In our given problem, P_0 = 2000 MW, and r = 1.3% = 0.013. Thus, the exponential growth function for the city's power rate is: P(t) = 2000 * (1 + 0.013)^t

Step 2: Finding the total energy used over four years

To find the total energy used by the city over four years (t=0 to t=4), we will calculate the definite integral of the previously found exponential growth function, P(t), with respect to time. By calculating this integral, we are summing up all the power used by the city during the given time period. The integral can be written as: Total Energy = ∫(2000 * (1 + 0.013)^t) dt from 0 to 4 To solve the integral, we'll perform a substitution: u = 1 + 0.013 = 1.013, du =0.013dt Thus, dt = du/0.013 Therefore, the integral becomes: Total Energy = (1/0.013) ∫(2000 * u^t) du from 0 to 4 Now we integrate with respect to u: Total Energy = (1/0.013) * (2000 * (u^(t+1))/(t+1)) from 0 to 4 Then, we plug in the original value of u (1.013): Total Energy = (1/0.013) * (2000 * (1.013^(t+1))/(t+1)) from 0 to 4 Finally, we evaluate the integral: Total Energy ≈ 8198.36 MW-yr

Step 3: Finding a function for the total energy used between t=0 and any future time t>0

We can find the general function for the total energy used (E(t)) between t=0 and any future time t>0 by taking the definite integral of the power function, P(t), with respect to time: E(t) = ∫(P(t)) dt = ∫(2000 * (1 + 0.013)^t) dt from 0 to t By using the same approach as in step 2, we can find the general function for the total energy E(t): E(t) = (1/0.013) * (2000 * (1.013^(t+1))/(t+1)) - (1/0.013) * (2000 * (1.013^(0+1))/(0+1)) After simplification, we get: E(t) ≈ 153846.15 * ((1.013^(t+1))/(t+1)) - 153846.15 So, the function E(t) gives the total energy used between t=0 and any future time t>0.

Key Concepts

Exponential Growth Function

To understand exponential growth functions, let’s start with a practical example. Imagine a city's electricity consumption starting at 2000 megawatts (MW) and increasing each year by a rate of 1.3%. This growth is not linear but rather exponential, meaning it accelerates over time.

In mathematical terms, the exponential growth function that models this scenario is given by:

• \( P(t) = P_0 \cdot (1 + r)^t \)

Here,

• \( P(t) \) is the power usage at time \( t \),
• \( P_0 \) is the initial power usage, which is 2000 MW,
• \( r \) is the growth rate, converted into decimal form, thus 0.013 for 1.3%,
• \( t \) represents the time in years.

This function effectively captures how the city's electricity demand scales over time, predicting future increases based on the present growth rate.

Definite Integral

The concept of a definite integral in calculus helps us find the total accumulation of quantities, such as energy consumption over a certain period. When dealing with functions like our exponential growth function for power rate, definite integrals play a key role.

To find out the total energy consumption over four years, we compute the definite integral of the function \( P(t) \) from \( t = 0 \) to \( t = 4 \):

• \( \int_{0}^{4} 2000 \cdot (1.013)^t \; dt \)

The result of this integral gives us the total energy consumed over that period in megawatt-years (MW-yr). The process involves calculating the area under the curve of \( P(t) \) from time zero to time four, capturing the sum total of all power used during this interval. Solving this integral provides a clear picture of energy used, which is approximately 8198.36 MW-yr.

Energy Consumption Calculation

Calculating energy consumption involves understanding how to use a derived function from calculus to model real-world scenarios. We need a function to determine energy usage at any given time in the future.

By setting up a function \( E(t) \), which represents the total energy used from the initial time \( t = 0 \) to any future time \( t \), we can adapt to future demands. The function is derived by performing the definite integral over the interval from zero up to time \( t \):

• \( E(t) = \int_{0}^{t} 2000 \cdot (1.013)^u \; du \)

This results in the following function:

• \( E(t) = \frac{153846.15 \cdot (1.013^{t+1})}{t+1} - 153846.15 \)

In practical terms, this expression estimates the total energy consumption at any point \( t \), allowing city planners and energy companies to anticipate and respond to changes in electricity demand efficiently.