Problem 65 - Algebra and Trigonometry

Question

Dimensions of a Structure A storage bin for corn consists of a cylindrical section made of wire mesh, surmounted by a conical tin roof, as shown in the figure. The height of the roof is one-third the height of the entire structure. If the total volume of the structure is 1400\(\pi \mathrm{ft}^{3}\) and its radius is 10 \(\mathrm{ft}\) , what is its height? \([\text { Hint: Use the volume formulas listed on }\) the inside front cover of this book.

Step-by-Step Solution

Answer

The height of the structure is approximately 20.65 ft.

Step 1: Define the Dimensions

Let the total height of the structure be denoted by \( h \). Then the height of the roof, being one-third of the total, is \( \frac{h}{3} \). Consequently, the height of the cylindrical section is \( h - \frac{h}{3} = \frac{2h}{3} \). The radius of both the cylindrical section and the conical roof is given as 10 ft.

Step 2: Calculate Volume of the Cylinder

The volume of a cylinder is given by \( V_{\text{cylinder}} = \pi r^2 h \). For our cylinder section, the volume is \[ V_{\text{cylinder}} = \pi \times 10^2 \times \frac{2h}{3} = \frac{2000\pi h}{3}. \]

Step 3: Calculate Volume of the Cone

The volume of a cone is given by \( V_{\text{cone}} = \frac{1}{3} \pi r^2 h \). For our conical roof, the volume is \[ V_{\text{cone}} = \frac{1}{3} \times \pi \times 10^2 \times \frac{h}{3} = \frac{100\pi h}{9}. \]

Step 4: Sum the Volumes

The total volume of the structure is the sum of the cylindrical section and the conical roof. Thus, \[ V_{\text{total}} = V_{\text{cylinder}} + V_{\text{cone}} = \frac{2000\pi h}{3} + \frac{100\pi h}{9}. \] This simplifies to \[ V_{\text{total}} = \pi h \left(\frac{2000}{3} + \frac{100}{9}\right) = \pi h \left(\frac{6000}{9} + \frac{100}{9}\right) = \frac{6100\pi h}{9}. \]

Step 5: Solve for Height

The total volume is given as 1400\(\pi\), so set \( \frac{6100\pi h}{9} = 1400\pi \) and solve for \( h \). Cancel out \( \pi \) from both sides and multiply through by 9 to get: \[ 6100h = 12600. \] Divide both sides by 6100 to solve for \( h \): \[ h = \frac{12600}{6100} = \frac{126\times 100}{61\times 100} = \frac{126}{61} \approx 20.65. \] Hence, the total height \( h \) is approximately 20.65 ft.

Key Concepts

Cylindrical Volume

A cylinder is a 3-dimensional shape with two parallel bases, both of which are circular. To find the volume of a cylinder, you use the formula: \( V_{\text{cylinder}} = \pi r^2 h \) where \( r \) is the radius of the base, and \( h \) is the height of the cylinder. This formula essentially helps you find out how much space is inside the cylinder.

It's important you remember:

• The base of the cylinder must be a circle, and \( \pi \) (Pi) is always approximately 3.14159.
• This formula applies only to cylinders; other shapes require different calculations.
• Measure height and radius using the same units to ensure consistency.

Knowing how to compute a cylinder's volume is handy in real life, especially in contexts such as storage or capacity estimations.

Conical Volume

A cone, much like a cylinder, also features a circular base and extends upward to form a peak. Calculating the volume of a cone involves using a slightly different formula: \( V_{\text{cone}} = \frac{1}{3} \pi r^2 h \) where \( r \) is the radius of the base, and \( h \) is the height from the base to the tip of the cone.

The cone's volume formula is derived from the cylinder's formula, emphasizing that a cone fills up only one-third of the cylinder volume if they share the same base and height.

• The multiplication by \(\frac{1}{3}\) reflects the cone's pointed shape, which encloses less space than a cylinder.
• The calculations are the same if the base and height are consistent units.
• Compared to cylindrical shapes, cones apply more in scenarios like funnels and roofs.

Understanding this formula is helpful for tasks such as calculating fills for tanks or other conical structures.

Volume Calculation

Volume calculation is essential for determining the space a 3-dimensional object occupies. Both cylindrical and conical shapes require specific formulas, but they can be combined when dealing with composite structures.

In the given exercise, the total volume of the structure is the sum of the volumes of both the cylindrical and conical sections. This requires:

• First, finding the separate volumes for each shape using their respective formulas.
• Adding those volumes to get a comprehensive total.
• Applying algebra to solve for unknown measurements such as height.

Remember that the key to successfully calculating volume lies in consistency within unit measurements and the appropriate application of each shape's formula. Mastering these concepts provides a strong foundation for not just geometrical problems, but real-world space planning and engineering as well.