Answer:
t = 16.6
Step-by-step explanation:
In the formula:
[tex]a = p(1 + r )^{t} [/tex]
A is the final amount, in this case is 11500 people. P is the initial amount, in this case is 6500 people. r is the growth rate, which in this case is 0.035 - we have to express it s a decimal, not a percent. And finally t is the time, which is the variable we want to find.
Let's clear t first and replace all these values at the end. To clear t we have to leave only the factor which exponent is t on one side of the equation. To do this we have to divide both sides by P:
[tex] \frac{a}{p} = \frac{p}{p} (1 + r) ^{t} \\ \frac{a}{p} = (1 + r)^{t} [/tex]
Now we have to use the following rule for the exponents and logarithms:
[tex] {a}^{x} = b[/tex]
Apply logarithms on both sides:
[tex]log( {a}^{x}) = log(b)[/tex]
By the exponents rule of logarithms
[tex]x \: log(a) = log(b)[/tex]
For this problem we have:
[tex]log \frac{a}{p} = log(1 + r)^{t} \\ log \frac{a}{p} =t \: . \: log(1 + r)[/tex]
Now we have to divide both sides by log(1+r) to clear t:
[tex]log \frac{a}{p} = t \: . \: \frac{log(1 + r)}{log(1 + r)} \\ t = \frac{log \: \frac{a}{p} }{log(1 + r)} [/tex]
And finally we just have to replace the values into this equation we found: A = 11500, P = 6500 and r = 0.035:
[tex]t = log \: \frac{11500}{6500} \\ log(1 + 0.035) \\ log \: \frac{23}{13} \\ t = log \: 1.035 \\ t = 16.6[/tex]
