Answer:
See Below.
Step-by-step explanation:
Part 1)
Since the coral structure has an initial volume of 1,200 cubic centimers, a = 1200.
Since it doubles every year, we can let b = 2.
Hence, our equation is:
[tex]y=1200(2)^t[/tex]
Where t is the time in years since the coral was measured.
Part 2)
Evaluate each value:
[tex]y(5)=1200(2)^5=38,400\text{ cm}^3[/tex]
When t = 5, or after five years, the volume of the coral structure is 38,400 cubic centimeters.
[tex]y(1)=1200(2)^1=2,400\text{ cm}^3[/tex]
When t = 1, or after one year, the volume of the coral structure is 2,400 cubic centimeters.
[tex]y(0)=1200(2)^0=1, 200\text{ cm}^3[/tex]
When t = 0, or during the initial year, the volume of the coral structure is 1,200 cubic centimeters.
And:
[tex]\displaystyle y(-1)=1200(2)^{-1}=\frac{1200}{2}=600\text{ cm}^3[/tex]
[tex]\displaystyle y(-2)=1200(2)^{-2}=\frac{1200}{4}=300\text{ cm}^3[/tex]
Part 3)
In this case, since positive values of t indicates the time in years since the coral was starting to be measured, negative values of t indicates the time in years before the coral was starting to be measured.
We acquired that y = 300 when t = -2. So, this means that two years before the coral was starting to be measured, the volume of the coral was 300 cubic centimeters.
Part 4)
We can set y = 37.5 and solve for t. Hence:
[tex]\displaystyle 37.5=1200(2)^t[/tex]
Divide both sides by 1200:
[tex]\displaystyle \frac{37.5}{1200}=\frac{375}{12000}=\frac{1}{32}=2^t[/tex]
Notice that:
[tex]\displaystyle \frac{1}{32}=\frac{1}{2^5}=2^{-5}=2^t[/tex]
Same bases must have the same exponents. Hence:
[tex]t=-5[/tex]
So, five years before the coral was being measured, the volume of the coral was 37.5 cubic centimeters.
Since the initial volume was 1,200 and it grows each year, it can only be 37.5 before the initial year, so our answer makes sense.
