the table shows an exponential relationship between the number of employees working for a business based on…

the table shows an exponential relationship between the number of employees working for a business based on the time, in years, since the founding of the business.\n\n| time since founding (years) | number of employees |\n| ---- | ---- |\n| 1 | 640 |\n| 2 | 800 |\n| 3 | 1,000 |\n| 4 | 1,250 |\n\ncreate an expression that can be used to determine the number of employees working for the business x years since the founding of the business.

the table shows an exponential relationship between the number of employees working for a business based on the time, in years, since the founding of the business.\n\n| time since founding (years) | number of employees |\n| ---- | ---- |\n| 1 | 640 |\n| 2 | 800 |\n| 3 | 1,000 |\n| 4 | 1,250 |\n\ncreate an expression that can be used to determine the number of employees working for the business x years since the founding of the business.

Answer

Explanation:

Step1: Identificar la forma general de una función exponencial

La forma general de una función exponencial es $y = a\cdot b^{x}$, donde $a$ es el valor inicial y $b$ es la tasa de crecimiento.

Step2: Encontrar el valor de $a$ y $b$

Tomando el primer par de valores $(x = 1,y = 640)$ y $(x = 2,y = 800)$. Sabemos que $y_1=a\cdot b^{1}$ y $y_2=a\cdot b^{2}$. Entonces $\frac{y_2}{y_1}=\frac{a\cdot b^{2}}{a\cdot b^{1}}=b$. Así, $b=\frac{800}{640}=\frac{5}{4}$. Sustituyendo $b = \frac{5}{4}$ y $x = 1,y = 640$ en $y=a\cdot b^{x}$, tenemos $640=a\cdot\frac{5}{4}$, entonces $a = 640\times\frac{4}{5}=512$.

Step3: Escribir la expresión

La expresión para determinar el número de empleados $y$ $x$ años después de la fundación del negocio es $y = 512\cdot(\frac{5}{4})^{x}$.

Answer:

$y = 512\cdot(\frac{5}{4})^{x}$