Problem 9 - Technical Mathematics with Calculus

Question

A cylindrical chimney has a round base with a 2.55 m outside diameter, a 2.00 m inside diameter, and a height of 7.54 m. Find the volume of masonry in the chimney.

Step-by-Step Solution

Answer

The volume of masonry in the chimney is \(\pi \times 7.54 m \(\left(1.275^2 - 1^2\right) m^2\)\) cubic meters.

Step 1: Calculate the volume of the outer cylinder

Use the formula for the volume of a cylinder, which is \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height. First, find the radius of the outer cylinder by halving the diameter: \(r_{outer} = \frac{2.55 m}{2} = 1.275 m\). Then calculate the volume of the outer cylinder: \(V_{outer} = \pi (1.275 m)^2 \times 7.54 m\).

Step 2: Calculate the volume of the inner cylinder

Similarly, find the volume of the inner void of the cylinder using the inside diameter. The radius of the inner cylinder is \(r_{inner} = \frac{2.00 m}{2} = 1.00 m\). Then calculate its volume: \(V_{inner} = \pi (1.00 m)^2 \times 7.54 m\).

Step 3: Calculate the volume of the masonry

Subtract the volume of the inner cylinder from the volume of the outer cylinder to find the volume of the masonry: \({V_{masonry} = V_{outer} - V_{inner}}\).

Step 4: Perform the subtraction to find the volume of masonry

After calculation, \(V_{masonry} \) works out to \(\pi (1.275 m)^2 \times 7.54 m - \pi (1.00 m)^2 \times 7.54 m\), which simplifies after factoring out the common terms, \(\pi \times 7.54 m \(\left(1.275^2 - 1^2\right) m^2\)\).

Step 5: Compute the numerical answer

Plug in the value of \(\pi\) and perform the arithmetic to get the final volume of the masonry in cubic meters.

Key Concepts

Volume of a Cylinder

Understanding the volume of a cylinder is a fundamental aspect of geometry that helps us determine how much space is inside a cylindrical object. Let's visualize a cylinder: it's similar to a soda can, having two flat ends and a curved surface. The formula to calculate the volume is \(V = \pi r^2 h\), with \(r\) being the radius of the circular base and \(h\) being the height of the cylinder. Just imagine filling your soda can with water; the amount of water it can hold is its volume.

For real-world problem-solving, like determining the material volume of a cylindrical chimney, you need to know these two measurements. It's important not to confuse diameter with radius; the radius is half the diameter. By plugging the appropriate values into this formula, we can calculate the space within any cylinder.

Cylinder

A cylinder is a three-dimensional shape that extends a circle into the third dimension. It consists of two parallel and congruent circles, known as the bases, which are connected by a curved surface at every point on the circles' circumferences. As with our chimney example, it's essential to distinguish between the outer cylinder, which represents the full size including the material and the inner hollow space, akin to how a pipe or a hollow column would be assessed.

It's also noteworthy to indicate that cylinders can be 'right' when the sides are perpendicular to the bases, or 'oblique' when they are not. Most exercises, including the chimney example, involve right cylinders, where the height forms a right angle with the base.

Subtracting Volumes

Subtracting volumes comes into play when we have a shape with an empty space inside that we need to discount from the total volume. Think of it as having a chocolate donut; you're interested in how much chocolate there is. If you measure both the whole donut and the hole, then subtract the volume of the hole, you’re left with the volume of just the chocolate.

Similarly, to find the volume of masonry in the chimney, we need to calculate the volume of the outside shape and then subtract the inside, hollow space. This ensures that we count the solid material only, which is crucial for practical applications like ensuring that we have enough materials or estimating the weight.

Mathematical Problem-Solving

Effective mathematical problem-solving involves both understanding concepts and applying formulas strategically. The key steps usually include identifying what is being asked, determining the known values, selecting the correct equations, and carrying out the calculations with precision.

Tips for Effective Problem-Solving

• Understand the problem: Read carefully to comprehend what is required.
• Organize the information: It helps to write down what you know and what you need to find out.
• Pick the best strategy: In the case of volume problems, it's often about using the right formula.
• Don't rush the calculations: Take your time with each step to avoid simple mistakes.
• Check your work: Review the process and the final answer to make sure it makes sense.

With these in mind, you can tackle mathematical problems effectively, building up your problem-solving toolkit one step at a time.