Answer:
Step-by-step explanation:
a) Firstly, we want to write the growth formula
We have the general form as:
![L\text{ = I\lparen1 + r\rparen}^t](https://img.qammunity.org/2023/formulas/mathematics/college/i977rvq2a4eu76trhs4x0x5stdf7mmgb5j.png)
where:
L is the estimated board feet of lumber
l is the is the initial estimated board feet of lumber (the initial estimated board feet of lumber in 2022)
r is the percentage rate of increase which is 9% (9/100 = 0.09)
t is the number of years to reach the estimated board feet of lumber
With respect to the question given, we have the formula as:
![\begin{gathered} L\text{ = 3,400,00\lparen1 + 0.09\rparen}^t \\ L\text{ =3,400,000\lparen1.09\rparen}^t \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/ocfyibjtplwjjqz06shh4pc06mhspl64md.png)
b) We want to get the value of t when L is 1 billion
Substituting the values, we have it that:
![\begin{gathered} 1000000000\text{ = 3400000\lparen1.09\rparen}^t \\ divide\text{ both sides by 3,400,000} \\ 294.12\text{ = 1.09}^t \\ ln\text{ 294.12 = tln 1.09t } \\ t\text{ = }\frac{ln\text{ 249.12}}{ln\text{ 1.09}}\text{ = 66 years} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/8dqqu22s1872lr2hbzolq44crqjjnt0nob.png)
In 66 years (2088) , the groove will reach one billion board feet of lumber