Question

There are currently 54 million cars in a certain country, decreasing exponentially by 6.8 % annually. How many years will it take for this country to have 46 million cars? Round to the nearest year.

Answer 1

16.7 years (not sure)

Explanation:

To solve this problem, we can use the formula for exponential decay:

y = y0 * (1 - r)^t

Where y is the number of cars after t years, y0 is the initial number of cars (54 million), r is the annual decrease rate as a decimal (6.8% = 0.068), and t is the number of years.

We want to find t when y = 46 million:

46 million = 54 million * (1 - 0.068)^t

Dividing both sides by 54 million:

46 million / 54 million = (1 - 0.068)^t

Taking the natural logarithm of both sides:

ln(46 million / 54 million) = ln((1 - 0.068)^t)

Using the logarithmic identity:

ln(46 million / 54 million) = t * ln(1 - 0.068)

Dividing both sides by ln(1 - 0.068):

t = ln(46 million / 54 million) / ln(1 - 0.068)

Using a calculator, t = approximately 16.7 years. Rounding to the nearest year, it will take 17 years for this country to have 46 million cars.