An HVAC technician is calculating the volume difference between two storage units, which is represented by the expression $(y + 1)^3 - 27y^3$. To factor this expression using the identity $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, which of the following correctly identifies the values for $a$ and $b$?

An inventory analyst is modeling the difference in cubic storage capacity between an updated shelving unit and an older design. The difference in volume is represented by the expression $(y + 1)^3 - 27y^3$. To begin factoring this expression using the difference of cubes formula, the analyst correctly identifies the second base term '$b$' as ${}27y$.

A logistics manager is analyzing the difference in storage volume between a standard cubic container and a new custom-designed model. The volume difference is represented by the expression $(y + 1)^3 - 27y^3$. To factor this expression, the manager uses the difference of cubes identity: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. Match each component of the manager's expression to its correct variable or product within the identity.

Identifying Bases in Cubic Volume Differences

A logistics supervisor at a distribution center is using the expression $(y + 1)^3 - {}27y^3$ to calculate the volume difference between two containers. To simplify this expression into its factored form using the difference of cubes identity, the supervisor follows a series of algebraic steps. Arrange these steps in the correct order.

A logistics coordinator is analyzing the volume difference between a standard cubic storage crate and a newly designed custom crate. The difference in volume is represented by the algebraic expression $(y + 1)^3 - 27y^3$. To factor this expression, the coordinator plans to use the difference of cubes identity, $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. If they set the first base term as $a = y + 1$, then the second base term $b$ is equal to ____.