Problem 27 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The volume of a cylindrical tank of radius \(r\) and height \(h\) is given by $$V=f(r, h)=\pi r^{2} h$$ Find the volume of a cylindrical tank of radius \(1.5 \mathrm{ft}\) and height \(4 \mathrm{ft}\).

Step-by-Step Solution

Answer

The volume of the cylindrical tank with radius \(1.5 ft\) and height \(4 ft\) is approximately \(28.274 ft^3\).

Step 1: We are given the radius \(r = 1.5 ft\) and the height \(h = 4 ft\). #Step 2: Substitute the given values into the formula#

We'll substitute the given values into the formula for the volume of a cylindrical tank: $$V = \pi (1.5)^2(4)$$ #Step 3: Calculate the volume#

Step 2: Now we have to simplify the equation: $$V = \pi (2.25)(4)$$ $$V = 9\pi$$ #Step 4: Convert the volume to a decimal number#

Finally, we convert the volume to a decimal number by multiplying by the value of pi (\(\approx 3.1416\)): $$V \approx 9(3.1416)$$ $$V \approx 28.274$$ Therefore, the volume of the cylindrical tank is approximately \(28.274 ft^3\).

Key Concepts

Cylinder Volume Calculation

Understanding how to calculate the volume of a cylinder is fundamental in applied mathematics, physics, and engineering. A cylinder's shape is characterized by its circular base and uniform height. The volume of a cylinder is the amount of space it occupies, measured in cubic units.

The general formula for the volume of a cylinder is expressed as \( V = \pi r^2 h \), where

• \( V \) is the volume,
• \( r \) is the radius of the circular base,
• \( h \) is the height of the cylinder, and
• \( \pi \) is a constant approximately equal to 3.1416.

While the formula itself might seem straightforward, understanding its components deeply helps in visualizing the problem. Each term in the formula corresponds to parts of the cylinder: the radius squared \( r^2 \) gives the area of its base, while multiplying this area by the height \( h \) extends it through the third dimension to find the total volume.

Substituting Values

Substituting values into an existing formula is a practice wherein specific numbers replace the variables in an equation. This step is essential for solving most mathematical problems including those in geometry. For example, if a cylindrical tank's radius and height are known, we can substitute these measurements into the volume formula for a cylinder to find its volume.

When substituting values, attention to detail is crucial. Make sure that the units are consistent and the values are squared or multiplied as the formula indicates. Using parentheses correctly helps to avoid any confusion during calculation, especially when multiple terms are involved. In a practical situation like calculating the volume of a cylindrical tank, substituting values correctly is vital to determine the precise space available for storing liquids or gases.

Geometry in Applied Mathematics

Geometry, a branch of mathematics, is significantly employed in practical fields like engineering, architecture, and physics. It allows us to describe and understand the shapes and sizes of objects we encounter in the real world. Applied mathematics involves using mathematical principles and techniques to solve real-world problems. Therefore, geometrical calculations become a key tool in this context.

Understanding the geometry of shapes like cylinders, cones, and spheres, for instance, is crucial when calculating volumes for storage, construction, or manufacturing purposes. It's not just about rigid formulas; it's about translating three-dimensional problems into solvable mathematical statements. This enables us to create designs, make informed decisions, and execute plans in various practical applications, reinforcing the importance of geometry in our everyday lives and in professional domains.