Question

the number of victims of a flu epidemic increases at a rate of 10% per week. there are currently 23 people infected. graph the exponential function that represents the number of victims after $t$ weeks.
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Explanation:

Step1: Define exponential growth formula

The general exponential growth function is $N(t) = N_0(1+r)^t$, where $N_0$ is the initial number, $r$ is the growth rate, and $t$ is time in weeks.

Step2: Substitute given values

Here, $N_0=23$, $r=0.10$. Substitute to get:
$N(t) = 23(1+0.10)^t = 23(1.10)^t$

Step3: Identify key points for graphing

• Initial point ($t=0$): $N(0)=23(1.10)^0=23$, so plot $(0, 23)$
• After 1 week ($t=1$): $N(1)=23(1.10)=25.3$, so plot $(1, 25.3)$
• After 2 weeks ($t=2$): $N(2)=23(1.10)^2=23(1.21)=27.83$, so plot $(2, 27.83)$
• After 3 weeks ($t=3$): $N(3)=23(1.10)^3=23(1.331)=30.613$, so plot $(3, 30.613)$

Step4: Sketch the curve

Connect the plotted points with a smooth, increasing curve that starts at $(0,23)$ and rises exponentially, never crossing the t-axis.

Answer:

The exponential function is $N(t)=23(1.10)^t$. The graph is an increasing exponential curve passing through points $(0,23)$, $(1,25.3)$, $(2,27.83)$, $(3,30.613)$, and continuing to rise as $t$ increases.