Question

the total number of people, p, who have been infected with a contagious virus t weeks after the epidemic began is given by the following formula

$p = \frac{1170}{1 + 4(0.5)^t}$

complete parts a through d below.

a. how many people originally had the virus?

originally 234 people were infected.

b. complete the table.

Explanation:

Step1: Substitute \(t = 0\) into the formula

When \(t = 0\), we have \(P=\frac{1170}{1 + 4(0.5)^{0}}\). Since \(a^{0}=1\) (\(a
eq0\)), then \((0.5)^{0}=1\). So \(P=\frac{1170}{1+4\times1}=\frac{1170}{5} = 234\).

Step2: Substitute \(t = 1\) into the formula

When \(t = 1\), \(P=\frac{1170}{1 + 4(0.5)^{1}}\). Since \((0.5)^{1}=0.5\), then \(P=\frac{1170}{1+4\times0.5}=\frac{1170}{1 + 2}=\frac{1170}{3}=390\).

Step3: Substitute \(t = 2\) into the formula

When \(t = 2\), \(P=\frac{1170}{1 + 4(0.5)^{2}}\). Since \((0.5)^{2}=0.25\), then \(P=\frac{1170}{1+4\times0.25}=\frac{1170}{1 + 1}=\frac{1170}{2}=585\).

Step4: Substitute \(t = 3\) into the formula

When \(t = 3\), \(P=\frac{1170}{1 + 4(0.5)^{3}}\). Since \((0.5)^{3}=0.125\), then \(P=\frac{1170}{1+4\times0.125}=\frac{1170}{1+0.5}=\frac{1170}{1.5}=780\).

Step5: Substitute \(t = 4\) into the formula

When \(t = 4\), \(P=\frac{1170}{1 + 4(0.5)^{4}}\). Since \((0.5)^{4}=0.0625\), then \(P=\frac{1170}{1+4\times0.0625}=\frac{1170}{1 + 0.25}=\frac{1170}{1.25}=936\).

Answer:

\(t\)	\(P\)
\(1\)	\(390\)
\(2\)	\(585\)
\(3\)	\(780\)
\(4\)	\(936\)