Question

have available and because viruses are able to mutate and change quickly, making it more difficult to contain them. you have been studying a new virus that causes people to break out in spots. suddenly, a colleague rushes into your office to inform you there is a confirmed outbreak of the virus in europe. the growth of the virus through a population is continuous (until it is somehow contained) at a rate of 3% per day. the current outbreak has 5 confirmed victims.

1. how many people will be infected with the virus on day 30?
2. create a model of the spread of the spotted virus in this region if it is not contained. to simplify your model slightly, consider the 5 victims as the number of victims on day 0.

table:
day(x) number of infected y
0 5
1 5.15
2 6.3045
3 5.4163635
30 12
equation: y = 5(1 + 0.03)^x

1. when does your model predict there will be 50 victims? show how you arrived at your answer.

y = 5(1.03)^x
50 = 5(1.03)^x
10 = (1.03)^x
ln(10) = xln(1.03)
x = ln(10)/ln(1.03)

1. will the number of days it will take the virus to claim 100 victims be double the number of days that it took to claim 50 victims? why or why not?
2. calculate the number of days it will take for the virus to claim 100 victims.
3. on what day will there be 150 victims?

now you have received a report of a mysterious illness that seems to turn the infected humans into mindless zombies has broken out in a major american city. since the hungry zombies prey upon innocent people, the outbreak grows continuously at a rate of 12% per day. the outbreak begins with 80 people.

1. how many zombies will there be after 5 days?

Explanation:

Step1: Identificar la fórmula de crecimiento exponencial

La fórmula de crecimiento exponencial es $y = a(1 + r)^t$, donde $a$ es el valor inicial, $r$ es la tasa de crecimiento y $t$ es el tiempo.

Step2: Resolver el primer problema

Dado $a = 5$, $r=0.03$ y $t = 30$. Sustituimos en la fórmula:
$y=5(1 + 0.03)^{30}$
$y = 5\times1.03^{30}$
Usando una calculadora, $1.03^{30}\approx2.427262$
$y=5\times2.427262 = 12.13631\approx12$

Step3: Resolver el tercer problema

Dado $y = 50$, $a = 5$ y $r = 0.03$. Sustituimos en la fórmula $y=a(1 + r)^t$:
$50=5(1 + 0.03)^t$
Dividimos ambos lados por 5:
$10=(1.03)^t$
Tomamos el logaritmo natural de ambos lados:
$\ln(10)=\ln(1.03^t)$
Usando la propiedad $\ln(a^b)=b\ln(a)$:
$\ln(10)=t\ln(1.03)$
$t=\frac{\ln(10)}{\ln(1.03)}\approx76.03\approx76$

Step4: Resolver el cuarto problema

Para $y = 100$, $a = 5$ y $r = 0.03$. Sustituimos en la fórmula:
$100=5(1.03)^t$
Dividimos por 5:
$20=(1.03)^t$
Tomamos logaritmo natural:
$\ln(20)=t\ln(1.03)$
$t=\frac{\ln(20)}{\ln(1.03)}\approx99.66\approx100$
El tiempo para 50 víctimas es aproximadamente 76 días y para 100 víctimas es aproximadamente 100 días. No es doble, porque el crecimiento exponencial no es lineal.

Step5: Resolver el quinto problema

Para $y = 100$, $a = 5$ y $r = 0.03$.
$100=5(1.03)^t$
$20=(1.03)^t$
$t=\frac{\ln(20)}{\ln(1.03)}\approx99.66\approx100$ días

Step6: Resolver el sexto problema

Para $y = 150$, $a = 5$ y $r = 0.03$.
$150=5(1.03)^t$
$30=(1.03)^t$
Tomamos logaritmo natural:
$\ln(30)=t\ln(1.03)$
$t=\frac{\ln(30)}{\ln(1.03)}\approx113.42\approx113$ días

Step7: Resolver el séptimo problema

Para el brote de zombis, $a = 80$, $r = 0.12$ y $t = 5$.
Usando la fórmula $y=a(1 + r)^t$:
$y=80(1 + 0.12)^5$
$y=80\times1.12^5$
$1.12^5=1.762342$
$y=80\times1.762342 = 140.98736\approx141$ zombis

Answer:

1. Aproximadamente 12 personas.
2. Aproximadamente 76 días.
3. No, porque el crecimiento es exponencial y no lineal.
4. Aproximadamente 100 días.
5. Aproximadamente 113 días.
6. Aproximadamente 141 zombis.