an equation for the depreciation of a car is given by y = a(1 - r)^t, where y = current value of the car, a…

an equation for the depreciation of a car is given by y = a(1 - r)^t, where y = current value of the car, a = original cost, r = rate of depreciation, and t = time, in years. the value of a car is half what it originally cost. the rate of depreciation is 10%. approximately how old is the car?\no 3.3 years\no 5.0 years\no 5.6 years\no 6.6 years

an equation for the depreciation of a car is given by y = a(1 - r)^t, where y = current value of the car, a = original cost, r = rate of depreciation, and t = time, in years. the value of a car is half what it originally cost. the rate of depreciation is 10%. approximately how old is the car?\no 3.3 years\no 5.0 years\no 5.6 years\no 6.6 years

Answer

Answer:

D. 6.6 years

Explanation:

Step1: Asigna valores a la ecuación

Dado que $y=\frac{A}{2}$ (el valor actual es la mitad del valor original) y $r = 0.1$ (10% expresado como decimal), la ecuación $y = A(1 - r)^t$ se convierte en $\frac{A}{2}=A(1 - 0.1)^t$.

Step2: Simplifica la ecuación

Dividimos ambos lados de la ecuación $\frac{A}{2}=A(0.9)^t$ por $A$ (suponiendo $A\neq0$), obteniendo $\frac{1}{2}=(0.9)^t$.

Step3: Aplica logaritmos

Tomamos el logaritmo en ambos lados: $\log(\frac{1}{2})=\log((0.9)^t)$. Usando la propiedad $\log(a^b)=b\log(a)$, tenemos $\log(\frac{1}{2}) = t\log(0.9)$.

Step4: Resuelve para $t$

$t=\frac{\log(\frac{1}{2})}{\log(0.9)}$. Sabiendo que $\log(\frac{1}{2})=-\log(2)\approx - 0.3010$ y $\log(0.9)\approx-0.0458$, entonces $t=\frac{- 0.3010}{-0.0458}\approx6.6$.