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.


a. Based on these figures, find an exponential growth function for the power (rate of electricity use) for the city.

Verified step by step guidance

1. Identify the initial value and the growth rate from the problem. The initial power consumption at time \(t=0\) is 2000 MW, and the rate of increase is 1.3% per year.
2. Express the growth rate as a decimal for use in the exponential growth formula. Since 1.3% = 0.013, this will be the rate of growth per year.
3. Recall the general form of an exponential growth function: \(P(t) = P_0 \times (1 + r)^t\), where \(P_0\) is the initial amount, \(r\) is the growth rate per time period, and \(t\) is the time in years.
4. Substitute the known values into the formula: \(P(t) = 2000 \times (1 + 0.013)^t\).
5. This function models the power consumption in megawatts at any year \(t\) after the start of the year (when \(t=0\)).

Key Concepts

Exponential Growth Function

An exponential growth function models quantities that increase at a rate proportional to their current value. It is generally expressed as P(t) = P_0 * e^(rt), where P_0 is the initial amount, r is the growth rate, and t is time. This function is ideal for representing continuous percentage increases over time.

Growth Rate and Percentage Increase

The growth rate represents how quickly a quantity increases, often given as a percentage per time unit. To use it in an exponential model, the percentage must be converted to a decimal (e.g., 1.3% = 0.013) and interpreted as a continuous rate of change, which influences the exponent in the growth function.

Initial Condition in Modeling

The initial condition specifies the starting value of the quantity at time zero, serving as the base for the growth function. In this problem, the initial power usage is 2000 MW at t=0, which sets the value of P_0 in the exponential model and anchors the function to real data.