Question

Three solid cubes of lead, each with edges 10 centimeters long, are melted together in a level, rectangular-shaped pan. The base of the pan has inside dimensions of 20 centimeters by 30 centimeters, and the pan is 15 centimeters deep. If the volume of the solid lead is approximately the same as the volume of the melted lead, approximately how many centimeters deep is the melted lead in the pan?

• A. 2.5
• B. 3
• C. 5
• D. 7.5
• E. 9

Hint:

In problems where a substance is melted and recast into a different shape, the core principle is always the conservation of volume. Equate the volume of the original shape(s) to the volume of the final shape and solve for the unknown dimension.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a volume conservation problem. The total volume of the lead does not change when it is melted and reshaped. We need to calculate the initial total volume of the lead cubes and then determine the height (depth) this volume would occupy in the rectangular pan.
Step 2: Key Formula or Approach:
\begin{itemize} \item Volume of a cube = \((\text{edge length})^3\) \item Volume of a rectangular prism (the melted lead in the pan) = length \(\times\) width \(\times\) depth \end{itemize} Step 3: Detailed Explanation:
1. Calculate the total volume of the lead.
Each lead cube has an edge of 10 cm.
Volume of one cube = \((10 \text{ cm})^3 = 1000 \text{ cm}^3\).
Since there are three cubes, the total volume of lead is:
\[ \text{Total Volume} = 3 \times 1000 \text{ cm}^3 = 3000 \text{ cm}^3 \]
2. Calculate the depth of the melted lead in the pan.
This total volume of lead is now in the rectangular pan. The volume of the melted lead can be expressed as:
\[ \text{Volume} = (\text{Base Area}) \times (\text{depth}) \]
The base of the pan has dimensions 20 cm by 30 cm.
Base Area = \(20 \text{ cm} \times 30 \text{ cm} = 600 \text{ cm}^2\).
We set the total volume of lead equal to the volume formula for the pan:
\[ 3000 \text{ cm}^3 = 600 \text{ cm}^2 \times \text{depth} \]
To find the depth, we rearrange the formula:
\[ \text{depth} = \frac{3000 \text{ cm}^3}{600 \text{ cm}^2} = \frac{30}{6} \text{ cm} = 5 \text{ cm} \]
(The 15 cm depth of the pan is extra information to show that the lead does not overflow, as 5 cm<15 cm).
Step 4: Final Answer:
The melted lead is 5 centimeters deep in the pan.