Answer :
a. The number of graks after 1.5 years are given as follows: 707.
b. The same amount of graks and of sprogs is after 18 years.
What is an exponential function?
The definition of an exponential function is presented as follows:
[tex]y = a(b)^{\frac{x}{n}}[/tex]
In which the terms of the function are listed as follows:
- a is the initial amount.
- b is the rate of change.
- n is the period the rate of change.
Then the functions are given as follows for this problem:
- Sprogs: [tex]y = 4000(2)^{\frac{x}{6}}[/tex]
- Graks: [tex]y = 500(2)^{\frac{x}{3}}[/tex]
After 1.5 years, the number of graks is given as follows:
y = 500 x 2^(0.5) = 707 graks.
The amounts will be the same when:
[tex]4000(2)^{\frac{x}{6}} = 500(2)^{\frac{x}{3}}[/tex]
[tex]\frac{(2)^{\frac{x}{3}}}{(2)^{\frac{x}{6}}} = \frac{4000}{500}[/tex]
[tex]2^{\left(\frac{x}{3} - \frac{x}{6}\right)} = 8[/tex]
[tex]2^{\left(\frac{x}{3} - \frac{x}{6}\right)} = 2^3[/tex]
Hence:
x/3 - x/6 = 3
x/6 = 3.
x = 18 years.
Missing Information
In the first question, it is after 1.5 years.
More can be learned about exponential functions at https://brainly.com/question/25537936
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