Question

1. dada una empresa que modela sus ingresos con una función exponencial: r(x)=1000(1.1)^x, donde x es el número de meses. a) ¿cuál es el crecimiento para x = 0 hasta x = 5? b) ¿qué tipo de crecimiento presenta? ¿es realista a largo plazo?

Explanation:

Step1: Recall exponential - growth formula

The given function is $R(x)=1000(1.1)^x$, which is in the form of an exponential - growth function $y = a(b)^x$, where $a = 1000$ and $b=1.1>1$.

Step2: Analyze the growth rate

The growth rate is determined by the base of the exponential function. For an exponential function $y = a(b)^x$, the growth rate is given by $(b - 1)\times100\%$. Here, $b = 1.1$, so the growth rate is $(1.1-1)\times100\%=10\%$.

Step3: Consider long - term behavior

As $x\to+\infty$ (long - term), since $b = 1.1>1$, the function $R(x)=1000(1.1)^x$ will grow without bound. In a real - world business context, this may not be realistic because there are limitations such as market saturation, competition, and resource constraints.

Answer:

a) The growth rate is $10\%$.
b) It is not realistic in the long - term because of market and resource limitations.