If you deposit $1000 in a savings account with an interest rate of r compounded annually, then the balance in the account after 3 years is given by the function B(c)=1000(1+r)^3, where r is written as a decimal. What is the formula for the interest rate, r; required to achieve a balance of B in the account after 3 years?

Our equation is as follows, B(c)=1000(1+r)^3. We are being asked to manipulate the equation and have it be r equals in order to solve for an interest rate when given a balance B. To solve the equation for the variable r, we must isolate r by completing order of operations or PEMDAS backwards.

SADMEP allows us to undo subtraction, addition, and etc in the correct order. We have no subtraction or addition outside of more complex operations. So we move to multiplication or division.

We divide both sides by 1000.

[tex]\frac{B}{1000} =\frac{1000(1+r)^{3} }{1000}[/tex]

We simplify the right side.

[tex]\frac{B}{1000} =(1+r)^{3}[/tex]

We need to now undo the exponent of 3 by using the inverse, a cube root.

[tex]\sqrt[3]{\frac{B}{1000}} =\sqrt[3]{(1+r)^{3}}[/tex]

We simplify the right side.

[tex]\sqrt[3]{\frac{B}{1000}} =(1+r)[/tex]

We subtract 1 to both sides to get r alone.

[tex]-1+\sqrt[3]{\frac{B}{1000}} =r[/tex]

For our last step, we simplify the denominator of the root because 10*10*10=1000.

[tex]-1+\frac{\sqrt[3]{b} }{10} =r[/tex]

This is answer choice b.