Solve each equation for the indicated variable. Round answers to two decimal places as necessary.
$\frac{\sqrt[3]{B}}{2}+2=-2$
Answer (1)
Subtract 2 from both sides: 2 3 B = − 4 .
Multiply both sides by 2: 3 B = − 8 .
Cube both sides: B = ( − 8 ) 3 = − 512 .
The solution is − 512 .
Explanation
Problem Analysis We are given the equation 2 3 B + 2 = − 2 and our goal is to solve for the variable B .
Isolating the Cube Root Term First, we need to isolate the term containing B . To do this, we subtract 2 from both sides of the equation: 2 3 B + 2 − 2 = − 2 − 2 2 3 B = − 4
Isolating the Cube Root Next, we multiply both sides of the equation by 2 to further isolate the cube root term: 2 × 2 3 B = 2 × ( − 4 ) 3 B = − 8
Solving for B Now, to solve for B , we need to eliminate the cube root. We do this by cubing both sides of the equation: ( 3 B ) 3 = ( − 8 ) 3 B = − 512
Final Answer Since the result is an integer, rounding to two decimal places is not necessary. Therefore, the solution is B = − 512 .
Examples
Imagine you are designing a refrigeration system and need to determine the volume of a coolant tank. The equation you solved is analogous to a simplified model where 'B' represents a parameter affecting the cooling efficiency. Solving for 'B' helps you optimize the tank size to achieve the desired cooling performance. This type of algebraic manipulation is crucial in engineering to relate design parameters and performance metrics, ensuring efficient and effective system design.
