Question

The volume of a certain substance is always directly proportional to its weight. If 48 cubic inches of the substance weigh 112 ounces, what is the volume, in cubic inches, of 63 ounces of this substance? $$\begin{array}{c}27\\ 36\\ 42\\ 64\\ 147\end{array}$$

Short answer

27 cubic inches.

Step-by-step solution

- Understand the problem

The volume (V) of the substance is directly proportional to its weight (W). This means we can describe the relationship as: $V=kW$, where $k$ is a constant of proportionality.

- Find the constant of proportionality

Given that 48 cubic inches of the substance weigh 112 ounces, we can substitute these values into the equation to find $k$. $$48=k\times 112$$ Solving for $k$, we get: $$k=\frac{48}{112}=\frac{3}{7}$$

- Set up the equation for unknown volume

Now that we have the constant of proportionality $k$, we can find the volume of 63 ounces of the substance. Using the proportionality constant, we set up the equation: $$V=\frac{3}{7}\times 63$$

- Solve for the unknown volume

Substitute 63 into the equation: $$V=\frac{3}{7}\times 63$$ Simplifying this, we get: $$V=27$$

Key concepts

Volume Calculation

Volume calculation for substances is a common problem in mathematics and science. Volume refers to the space that a substance occupies. In the context of this problem, the volume is given in cubic inches. When calculating volume, it's essential to understand the detailed relationship between the measure you're given (like weight) and volume. For instance, if you know the volume of 48 cubic inches, it can be used with direct proportionality to find another unknown volume based on weight. You start by establishing the relationship using a formula or equation that connects your given data to what you need to find. This involves critical steps like identifying given values, recognizing proportionality, and appropriately substituting into equations.

Constant of Proportionality

The constant of proportionality, often denoted by the symbol $k$, is a crucial concept in problems involving direct proportionality. It represents the specific ratio or relationship between two varying quantities. In this exercise, the volume ($V$) is directly proportional to the weight ($W$). This relationship is written as: $$V=kW$$ To find the constant of proportionality, you need initial paired values of volume and weight. Here, given 48 cubic inches and 112 ounces, you substitute them into the equation to get: $$48=k\times 112$$ By solving for $k$, you get: $$k=\frac{48}{112}=\frac{3}{7}$$ This constant $(\frac{3}{7})$ is then used to calculate unknown values in similar relationships. Understanding this constant simplifies complex calculations by allowing direct multiplication with the given quantity to find the needed value.

Weight-Volume Relationship

Weight-volume relationship is about understanding how the weight of a substance affects its volume, especially when they are directly proportional. In simpler terms, as the weight of something increases, so does its volume, and this is uniformly scaled by a constant factor. This problem highlights this with the volume and weight of a substance. For instance, if 48 cubic inches of the substance weigh 112 ounces, and you know the constant of proportionality is $\frac{3}{7}$, then you can find any other corresponding volume for a different weight using this relationship.

To find the volume corresponding to 63 ounces, use the established proportional equation: $$V=\frac{3}{7}\times 63$$ Substituting 63 for weight (W), and simplifying: $$V=27$$ This shows that 63 ounces of this substance would occupy 27 cubic inches of volume. Thus, by understanding and applying the weight-volume relationship, you can accurately determine volumes for varying weights using the constant of proportionality.