Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.


b. Find the total energy (in MW-yr) used by the city over four full years beginning at t=0.

Verified step by step guidance

1. Identify the given information: the initial rate of electricity consumption at time \(t=0\) is \(R_0 = 2000\) MW, and the rate increases at a rate of 1.3% per year. This means the rate of consumption grows exponentially.
2. Express the rate of electricity consumption as a function of time \(t\) (in years). Since the rate increases by 1.3% per year, the rate function is \(R(t) = 2000 \times (1.013)^t\) MW.
3. To find the total energy used over four years, we need to integrate the rate function over the interval from \(t=0\) to \(t=4\). The total energy \(E\) is given by the integral \(E = \int_0^4 R(t) \, dt = \int_0^4 2000 \times (1.013)^t \, dt\).
4. Set up the integral for the exponential function. Recall that the integral of \(a^t\) with respect to \(t\) is \(\frac{a^t}{\ln(a)}\), where \(a > 0\) and \(a \neq 1\). So, the integral becomes \$2000 \times \int_0^4 (1.013)^t \, dt = 2000 \times \left[ \frac{(1.013)^t}{\ln(1.013)} \right]_0^4$.
5. Evaluate the definite integral by substituting the limits \(t=4\) and \(t=0\) into the antiderivative expression, then subtract the lower limit value from the upper limit value to find the total energy consumed over the four years.

Key Concepts

Exponential Growth

Exponential growth describes a quantity that increases at a rate proportional to its current value. In this problem, the electricity usage rate grows by 1.3% per year, meaning the rate at time t can be modeled as an exponential function, R(t) = R_0 * e^(kt), where k is the growth rate.

Integration to Find Total Accumulated Quantity

To find the total energy used over a time interval, we integrate the rate function over that period. Integration sums the instantaneous rates over time, giving the total accumulated energy consumption in MW-years.

Converting Percentage Growth to Continuous Growth Rate

A percentage growth rate given per year (1.3%) can be converted to a continuous growth rate k by using k = ln(1 + r), where r is the decimal form of the percentage. This allows modeling the growth with the exponential function e^(kt) for continuous compounding.