Question

a country commits to decreasing spending for infrastructure in various ways at a rate of 30% per year. at the time of the announcement, the country is spending $12 billion per year. which graph models the amount of infrastructure spending for future years? (graph with a curve starting at (0, 30) and decreasing)

Explanation:

Step1: Identify the model type

This is an exponential decay problem. The general formula for exponential decay is $y = a(1 - r)^t$, where $a$ is the initial amount, $r$ is the rate of decay, and $t$ is time. Here, $a = 12$ billion, $r = 0.3$ (30% decay rate). So the formula is $y = 12(1 - 0.3)^t=12(0.7)^t$.

Step2: Analyze the y - intercept

The y - intercept of an exponential function $y = a(b)^t$ occurs at $t = 0$. Substituting $t = 0$ into our function, we get $y=12(0.7)^0 = 12$. So the graph should pass through the point $(0,12)$. The given graph in the problem has a y - intercept of $(0,30)$, which does not match our initial value of 12. But since we are to identify the type of graph (exponential decay), we know that the graph of exponential decay is a curve that starts at the initial value (on the y - axis when $t = 0$) and decreases rapidly at first and then more slowly as $t$ increases. The general shape of an exponential decay function $y=a(1 - r)^t$ (with $0<1 - r<1$) is a decreasing curve, concave up, with a y - intercept at $(0,a)$. In our case, $a = 12$, so the correct graph should have a y - intercept at $(0,12)$ and be a decreasing exponential curve.

Answer:

The graph should be an exponential decay curve with a y - intercept at $(0,12)$ (representing the initial spending of $12$ billion) and decreasing over time according to the function $y = 12(0.7)^t$. The given graph in the problem has a y - intercept of $(0,30)$ which is incorrect for this problem, but the correct graph will have the shape of an exponential decay (a decreasing curve, concave up) with $y(0)=12$.