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Explanation:

Step1: Set up the compound - growth formula

The compound - growth formula is $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the initial amount, $r$ is the annual growth rate, and $t$ is the number of years. We know that $P = 4000$, $A=14000000$, and $t = 40$. Substituting these values into the formula gives $14000000=4000(1 + r)^{40}$.

Step2: Solve for $(1 + r)^{40}$

First, divide both sides of the equation $14000000 = 4000(1 + r)^{40}$ by $4000$. We get $\frac{14000000}{4000}=(1 + r)^{40}$, so $(1 + r)^{40}=3500$.

Step3: Solve for $1 + r$

Take the 40th - root of both sides. $1 + r = 3500^{\frac{1}{40}}$. Using a calculator, $3500^{\frac{1}{40}}\approx1.214$.

Step4: Solve for $r$

Subtract 1 from both sides to find $r$. $r=3500^{\frac{1}{40}}-1\approx1.214 - 1=0.214$.

Step5: Convert $r$ to a percentage

Multiply $r$ by 100 to get the percentage. $r\times100 = 21.4%$.

Answer:

$21.4$