Question

In​ 2000, a company had 1147 stores nationwide. By​ 2002, this total had grown to 1542. If the number of stores continues to grow exponentially at the same​ rate, how many stores will there be in​ 2016?

Answer 1

12226 stores will be there in 2016

Solution:

The growth function is given as:

$y = a(1+r)^t$

Where,

y is the future value

a is the initial value

r is the growth rate

t is the number of years

From given,

In​ 2000, a company had 1147 stores nationwide

By​ 2002, this total had grown to 1542

Therefore,

y = 1542

a = 1147

t = 2000 to 2002 = 2 years

r = ?

Substituting we get,

$1542 = 1147(1+r)^2\\\\(1+r)^2 = \frac{1542}{1147}\\\\(1+r)^2 = 1.34437$

Taking square root of both sides

$1+ r = 1.1594\\\\r = 1.1594 - 1\\\\r = 0.1594$

If the number of stores continues to grow exponentially at the same​ rate, how many stores will there be in​ 2016?

Therefore,

y = ?

a = 1147

r = 0.1594

t = 2000 to 2016 = 16 years

Substituting we get,

$y = 1147(1 + 0.1594)^{16}\\\\y = 1147(1.1594)^{16}\\\\y = 1147 \times 10.6593\\\\y = 12226.3311 \approx 12226$

Thus 12226 stores will be there in 2016