Answer:
Final answer is approx 6.644 years.
Step-by-step explanation:
Given that a company sold 5.75 million motorcycles and 3.5 million cars in the year 2010. The growth in the sale of motorcycles is 16% every year and that of cars is 25% every year.
So we can use growth formula:
[tex]A=P\left(1+r\right)^t[/tex]
Then we get equation for motorcycles and cars as:
[tex]A=5.75\left(1+0.16\right)^t[/tex]
[tex]A=3.5\left(1+0.25\right)^t[/tex]
Now we need to find about when the sale of cars will be more than the sale of motorcycles. So we get:
[tex]3.5\left(1+0.25\right)^t>5.75\left(1+0.16\right)^t[/tex]
[tex]3.5\left(1.25\right)^t>5.75\left(1.16\right)^t[/tex]
[tex]3.5\left(1.25\right)^t>5.75\left(1.16\right)^t[/tex]
[tex]\frac{\left(1.25\right)^t}{\left(1.16\right)^t}>\frac{5.75}{3.5}[/tex]
[tex]\left(\frac{1.25}{1.16}\right)^t>1.64285714286[/tex]
[tex]t\cdot\ln\left(\frac{1.25}{1.16}\right)>\ln\left(1.64285714286\right)[/tex]
[tex]t>\frac{\ln\left(1.64285714286\right)}{\ln\left(\frac{1.25}{1.16}\right)}[/tex]
[tex]t>6.6436473051[/tex]
Hence final answer is approx 6.644 years.
