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

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?\n3.3 years\n5.0 years\n5.6 years\n6.6 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?\n3.3 years\n5.0 years\n5.6 years\n6.6 years

Answer

Explanation:

Step1: Substitute given values into formula

Given $y=\frac{A}{2}$, $r = 0.1$. Substitute into $y=A(1 - r)^t$, we get $\frac{A}{2}=A(1 - 0.1)^t$.

Step2: Simplify the equation

Divide both sides of $\frac{A}{2}=A(0.9)^t$ by $A$ (since $A\neq0$), we have $\frac{1}{2}=(0.9)^t$.

Step3: Take the natural - logarithm of both sides

$\ln(\frac{1}{2})=\ln((0.9)^t)$. Using the property $\ln(a^b)=b\ln(a)$, we get $\ln(\frac{1}{2}) = t\ln(0.9)$.

Step4: Solve for $t$

$t=\frac{\ln(\frac{1}{2})}{\ln(0.9)}=\frac{-\ln(2)}{\ln(0.9)}$. Since $\ln(2)\approx0.693$ and $\ln(0.9)\approx - 0.105$, then $t=\frac{- 0.693}{-0.105}\approx6.6$.

Answer:

6.6 years