Problem 97 - Prealgebra

Question

Sue has decided to build a circular fish pond near her patio. She wants it to be 15 feet in diameter and \(1.5\) feet deep. What is the volume of water it will hold? Use \(\pi \approx 3.14\). Hint: The volume of a cylinder is given by the formula \(V=\pi r^{2} h\), which is the area of the circular base times the height of the cylinder.

Step-by-Step Solution

Answer

The pond will hold approximately 264.94 cubic feet of water.

Step 1: Understand the Formula

The volume of a cylinder is calculated by multiplying the area of the circular base by the height of the cylinder. The formula is given by \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height of the cylinder (or depth in the case of the pond).

Step 2: Find the Radius

The diameter of the pond is given as 15 feet. The radius \( r \) can be found by dividing the diameter by 2. So, \( r = \frac{15}{2} = 7.5 \) feet.

Step 3: Identify the Height

The depth of the pond, which will be the height of the cylinder, is given as \( 1.5 \) feet. So, \( h = 1.5 \) feet.

Step 4: Substitute Values Into Formula

Substitute the radius \( r = 7.5 \) feet and the height \( h = 1.5 \) feet into the volume formula: \( V = \pi (7.5)^2 (1.5) \).

Step 5: Calculate the Volume

First, calculate \( (7.5)^2 = 56.25 \) square feet. Then, multiply by \( 1.5 \) to get \( 56.25 \times 1.5 = 84.375 \) cubic feet. Finally, multiply by \( \pi \approx 3.14 \) to find the volume: \( 84.375 \times 3.14 \approx 264.9375 \) cubic feet.

Key Concepts

Understanding Geometry in Cylinder Volume Calculation

When we talk about calculating the volume of a cylinder, we're diving into the world of geometry. Geometry is the branch of mathematics that deals with shapes, sizes, and the properties of space. In this exercise, we're focusing on a specific geometric shape: the cylinder.

A cylinder is a 3D shape with two parallel circular bases connected by a curved surface. Think of it as a soup can or, in this exercise, a fish pond. Understanding the structure of a cylinder is crucial because we need to know how to measure it to calculate its volume.

• The base of the cylinder is a circle, and its size is determined by its radius or diameter.
• The height or depth of the cylinder is the straight line distance between the two bases.

In our problem, the fish pond is a type of cylinder, and we're interested in calculating how much water it can hold, which involves understanding its volume.

Applying Mathematical Formulas to Solve Problems

In mathematics, formulas are like recipes. They guide us on how to solve specific types of problems. The formula for the volume of a cylinder is one such tool. It gives us a systematic way to find the volume of any cylindrical object, like Sue's fish pond.

The volume formula for a cylinder is:\[ V = \pi r^2 h \]Here:

• \( V \) stands for volume,
• \( \pi \) (approximately 3.14) is a mathematical constant,
• \( r \) is the radius of the circular base,
• \( h \) is the height or depth of the cylinder.

To use this formula, it's crucial to correctly substitute the values for \( r \) and \( h \), as demonstrated in our problem. Once you have the values substituted, you perform the arithmetic to get the answer. This methodical approach is what makes mathematical formulas powerful and universally applicable.

Developing Problem-Solving Skills through Practice

Problem-solving is one of the most valuable skills you can develop through mathematical exercises. By working through problems like this one, you're not just learning about geometry or how to apply formulas; you're honing your ability to tackle various challenges.

Here's how you can further enhance your problem-solving skills:

• Break the problem down into smaller, manageable steps, as seen in the original solution. Understand each step thoroughly before moving to the next.
• Identify the information given and determine what you need to find. In this case, you have the diameter and depth and need to find the volume.
• Apply the appropriate formula or method. Be meticulous in doing calculations and don't hesitate to re-check your computations.
• Practice consistently. The more problems you solve, the more intuitive the process becomes.

This methodology not only aids in solving geometric problems but is beneficial across various disciplines, making you a more efficient thinker and problem-solver in general.