Problem 96 - Elementary Algebra

Question

Problem: Find the volume of a cylinder with a diameter of 14 in. and a height of \(2 \mathrm{ft}\). Write the answer in cubic inches. Use \(\pi \approx 3.14\). Incorrect Answer: \(V=\pi r^{2} h\) $$ \begin{aligned} &V=(3.14)(7 \text { in. })^{2}(2 \mathrm{ft}) \\ &V=(3.14)\left(49 \text { in. }^{2}\right)(2 \mathrm{ft}) \\ &V=307.72 \text { in. }^{3} \end{aligned} $$

Step-by-Step Solution

Answer

The volume of the cylinder is 3694.24 cubic inches.

Step 1: Convert Height to Inches

The height of the cylinder is given as 2 feet. Convert this into inches by using the conversion factor (1 foot = 12 inches). So, 2 feet is equal to \[2 \times 12 = 24 \text{ inches}\]

Step 2: Identify Radius

The diameter of the cylinder is 14 inches. The radius is half of the diameter. Hence, the radius is \[\frac{14}{2} = 7 \text{ inches}\]

Step 3: Write the Formula for Volume of Cylinder

The formula for the volume of a cylinder is \[V = \pi r^2 h\], where \(r\) is the radius and \(h\) is the height.

Step 4: Substitute Values into the Formula

Substitute the radius \(r = 7 \text{ inches}\) and height \(h = 24 \text{ inches}\) into the formula: \[V = 3.14 \times (7)^2 \times 24\]

Step 5: Calculate the Volume

First calculate \((7)^2\), which is 49. Then multiply the values: \[V = 3.14 \times 49 \times 24\] \[V = 3.14 \times 1176\] \[V = 3694.24 \text{ in}^3\]

Key Concepts

Geometry

Geometry is a branch of mathematics that studies the sizes, shapes, and properties of figures and spaces. When dealing with solid figures like a cylinder, it's essential to understand the different parts it consists of.
A cylinder is a three-dimensional shape with two parallel circular bases and a curved surface connecting them. The key measurements of a cylinder are its diameter, radius, and height. The diameter is the distance across the circular base through the center, while the radius is half of this distance. The height is the distance between the two bases. To solve geometry problems involving cylinders, these measurements are crucial.
In our problem, we’re given the diameter and height, and we need to find the volume – a concept that measures the amount of space inside the cylinder.

Cylindrical Volume Formula

To find the volume of a cylinder, we use a specific formula:
where: * \(V\) is the volume * \(\pi\) is a constant approximately equal to \(3.14\) * \(r\) is the radius of the circular base * \(h\) is the height of the cylinder To effectively use this formula, make sure that all measurements are in the same units. This ensures accurate results.

Unit Conversion

In some geometry problems, like the one provided, you may need to convert units to ensure all measurements are in the same unit for accurate calculation. Conversion is especially important when measurements are given in different scales.
The problem gives the cylinder height in feet and radius in inches. To find the volume accurately, we first convert the height from feet to inches using the conversion factor: \[1 \text{ foot} = 12 \text{ inches}\].You multiply the height in feet by 12
So, for 2 feet, the conversion is:
\begin{aligned} 2 \text{ feet} \ \times 12 \text{ inches/foot} = \ 24 \text{ inches} \end{aligned}\Now with the height also in inches, you can proceed to use the formula accurately.
\ul\ Key points in Unit Conversion: *Always ensure the units match before calculating * Use appropriate conversion factors for accurate measurements * Double-check calculations to avoid errors that might arise from incorrect unit conversion.