Problem 7 - Elementary Algebra Within Reach

Question

Geometry An Olympic-size swimming pool in the shape of a rectangular solid has a volume of 3125 cubic meters, a length of 50 meters, and a width of 25 meters. What is the depth of the pool?

Step-by-Step Solution

Answer

By performing the calculation, we find that the depth of the pool is 2.5 meters.

Step 1: Understanding the concept of volume for a rectangular solid

The volume of any rectangular solid can be calculated using the formula \( Volume = length * width * depth \). Here, we know the volume, length, and width and we need to find the depth.

Step 2: Plugging in the known values in the volume formula

Let's substitute the given values into the formula: \( 3125 = 50 * 25 * depth \) which simplifies to \( 3125 = 1250 * depth \).

Step 3: Calculating the Depth

Now, to find the value for the depth, we need to isolate it on one side of the equation. To do this, we divide both sides of the equation by 1250: \( depth = 3125 \div 1250 \).

Key Concepts

Calculating Depth

Calculating the depth of a rectangular solid, such as an Olympic-size swimming pool, starts with understanding the volume formula. The volume of a rectangular solid can be determined using the equation: \[ Volume = ext{length} \times ext{width} \times ext{depth} \] In the case of our swimming pool problem, we are provided with the volume, length, and width. Our goal is to find the depth.

• Volume = 3125 cubic meters
• Length = 50 meters
• Width = 25 meters

We put these values into our formula, replacing the depth with a variable we need to solve for. By substituting, the equation becomes: \[ 3125 = 50 \times 25 \times ext{depth} \] This allows us to focus on solving for depth by simplifying the equation thereafter.

Algebraic Manipulation

Algebraic manipulation is a critical step in solving equations for unknown variables. In the pool example, once you substitute known values into the volume formula, you have: \[ 3125 = 1250 \times ext{depth} \] The aim is to isolate the variable representing depth on one side of the equation, making it easier to solve. This involves performing operations to both sides of the equation. Here’s how it's done:

• First, identify the operation connecting the known values to the variable (in this case, multiplication)
• To isolate depth, divide both sides of the equation by 1250

This results in: \[ ext{depth} = \frac{3125}{1250} \] By calculating the right side, you’ll find that the depth equals 2.5 meters. This algebraic approach is crucial as it applies to many geometry problems, allowing you to solve for missing parts in a straightforward manner.

Rectangular Solid Geometry

Rectangular solids are three-dimensional shapes with six rectangular faces, such as boxes or swimming pools. Understanding these solid geometries involves knowing specific components such as length, width, and depth (or height). These are essential in calculating the volume of the shape.

• A rectangular solid's volume is calculated with the formula: \[ V = l \times w \times h \]
• Volume tells us how much space the solid occupies in a three-dimensional space
• Knowing any three out of the four measurements (volume, length, width, depth) allows you to solve for the unknown

This three-dimensional understanding can be applied widely, not just in calculating pools, but in any scenario involving regular box-shaped structures. Mastering rectangular solid geometry makes you better equipped to solve real-world problems related to volume and space.