Problem 31 - Algebra Form and Function

Question

A city's electricity consumption, \(E\), in gigawatt-hours per year, is given by $$ E=\frac{0.15}{p^{3 / 2}} $$ where \(p\) is the price in dollars per kilowatt-hour charged. (a) Is \(E\) a power function of \(p ?\) If so, identify the exponent and the constant of proportionality. (b) What is the electricity consumption at a price of \(\$ 0.16\) per kilowatt- hour? At a price of \(\$ 0.25\) per kilowatt hour? Explain the change in electricity consumption in algebraic terms.

Step-by-Step Solution

Answer

#tag_title# (Step 5: Simplify the ratio expression) #tag_content#To simplify the ratio of the electricity consumption at the two prices, we can cancel out the constants and simplify the expression: $$ \frac{E(p_1)}{E(p_2)} = \frac{\frac{0.15}{(0.16)^{3 / 2}}}{\frac{0.15}{(0.25)^{3 / 2}}} = \frac{(0.25)^{3 / 2}}{(0.16)^{3 / 2}} $$ This ratio indicates the change in electricity consumption when the price changes from \(0.16\) to \(0.25\) per kilowatt-hour. If this ratio is greater than 1, then the electricity consumption increases; if it is less than 1, the consumption decreases. Overall, the change in electricity consumption can be described algebraically by the simplified expression: $$ \frac{E(p_1)}{E(p_2)} = \frac{(0.25)^{3 / 2}}{(0.16)^{3 / 2}} $$

Step 1: (Step 1: Determine if E is a power function of p)

A power function has the form \(f(x) = kx^n\), where k is the constant of proportionality and n is the exponent. In our case, we are given the function: $$ E=\frac{0.15}{p^{3 / 2}} $$ We can rewrite this function to follow the power function form by multiplying the denominator to the other side: $$ E \cdot p^{3/2} = 0.15 $$ Since this equation meets the criteria for the power function, we can say \(E\) is a power function of \(p\).

Step 2: (Step 2: Identify the exponent and the constant of proportionality)

Now that we know E is a power function of p, we can identify the exponent and the constant of proportionality by comparing it to the general power function form, \(f(x) = kx^n\). The given function can be rewritten as: $$ E = 0.15 \cdot p^{-(3/2)} $$ Comparing it to the general power function form, we can identify the values: Constant of proportionality, k = \(0.15\), Exponent, n = \(-(3/2)\)

Step 3: (Step 3: Calculate electricity consumption at \(0.16 and \)0.25 per kilowatt-hour)

We are given the electricity consumption function: $$ E=\frac{0.15}{p^{3 / 2}} $$ We will plug in the values for p and calculate the corresponding electricity consumption, E. For \(p = 0.16\): $$ E=\frac{0.15}{(0.16)^{3 / 2}} $$ For \(p = 0.25\): $$ E=\frac{0.15}{(0.25)^{3 / 2}} $$

Step 4: (Step 4: Compare the change in electricity consumption at the two prices)

To explain the change in electricity consumption as the price changes from \(p_1=0.16\) to \(p_2=0.25\) in algebraic terms, we can compare the ratio of the electricity consumption at the two prices. We have already found the expression for E at \(p_1=0.16\) and \(p_2=0.25\). Now, let's find the ratio of the electricity consumption at the two prices: $$ \frac{E(p_1)}{E(p_2)} = \frac{\frac{0.15}{(0.16)^{3 / 2}}}{\frac{0.15}{(0.25)^{3 / 2}}} $$ We can simplify this expression to understand how the electricity consumption is changing as the price changes from \(0.16 to \)0.25 per kilowatt-hour.

Key Concepts

Exponent

In mathematics, an exponent refers to a number that signifies how many times a particular base number is multiplied by itself. It is a critical part of understanding power functions, where you often see expressions like\(f(x) = kx^n\), with \(n\) as the exponent.

In our exercise, the function given is\(E = \frac{0.15}{p^{3/2}}\). To match the typical power function format, this can be rewritten as \(E = 0.15 \cdot p^{-(3/2)}\).

This reveals the exponent is \(-\frac{3}{2}\), or \(-1.5\). A negative exponent signifies that the base number, in this case \(p\), is in the denominator. This means, as the price \(p\) increases, the electricity consumption \(E\) decreases, illustrating an inverse relationship.

Understanding exponents help you comprehend how values scale in power functions. They highlight how sensitive consumption is to changes in price, especially when they are negative, as they reflect a decrease in the response variable when the input increases.

Constant of Proportionality

The constant of proportionality in a power function describes the fixed factor by which the base variable is multiplied. It is represented by \(k\) in the power function equation\(f(x) = kx^n\).

In the exercise provided, the constant of proportionality is 0.15. This value acts as the scaling factor in our power function\(E = 0.15 \cdot p^{-(3/2)}\).

What does this mean practically? It's the baseline multiplier of the price factor. So, even as the price \(p\) varies, this constant ensures that the electricity consumption \(E\) maintains a consistent relationship with \(p\) throughout all price changes.

The constant is an important component as it gives the function its overall magnitude. In real-world contexts such as ours with electricity consumption, it reflects factors like base energy requirements or fixed costs that don't change regardless of pricing.

Electricity Consumption

Electricity consumption, in this context, refers to the amount of electrical energy used within a specified timeframe, measured in gigawatt-hours per year. The exercise models this with respect to the price of electricity per kilowatt-hour, highlighting how sensitive consumption might be to price changes.

With the relationship described by \(E=\frac{0.15}{p^{3/2}}\), the equation shows an inverse power function relation between electricity usage \(E\) and price \(p\). When the price of electricity increases, the consumption decreases as the negative exponent \(n = -\frac{3}{2}\) implies.

Consider practical calculations:

• At \(p = 0.16\), use the formula to calculate \(E\) which results in a specific consumption level.
• At \(p = 0.25\), the consumption \(E\) is recalculated, and observing both results, you see a lower consumption figure.

This decrease represents a typical consumer behavior model where a rise in prices tends to lower consumption, reflecting budget constraints or energy-saving measures. By understanding this equation, students can grasp the significant impact of electricity pricing on usage patterns.