Problem 44 - Algebra for College Students

Question

Marina City in Chicago is a complex of two residential towers that resemble corncobs. Each tower has a concrete cylindrical core with a 35 -ft diameter and is \(588 \mathrm{ft}\) tall. Find the volume of the core of one of the towers to the nearest whole number.

Step-by-Step Solution

Answer

The volume of the core of one tower is approximately 565,486 cubic feet.

Step 1: Identify the Given Values

The diameter of the cylindrical core is given as 35 feet, and the height is 588 feet. Use these values for further calculations.

Step 2: Calculate the Radius

The radius is half of the diameter. Therefore, the radius is \ \( r = \frac{35}{2} = 17.5 \ \text{ft} \).

Step 3: Use the Volume Formula for a Cylinder

The formula to calculate the volume of a cylinder is \ \(V = \pi r^2 h \). Substitute the values into the formula.

Step 4: Compute the Volume

Substitute \ \( r = 17.5 \ \text{ft} \) and \ \( h = 588 \ \text{ft} \) into the formula: \ V = \pi (17.5)^2 (588) \ Calculate this to find the volume.

Step 5: Simplify and Round to the Nearest Whole Number

First, calculate \ \( (17.5)^2 = 306.25 \). Then multiply \ \( 306.25 \) by \ \( 588 \) and \ \( \pi \). Approximate \ \( \pi \) as 3.1416, making the volume \ V = 3.1416 \times 306.25 \times 588 \ Finally, round the result to the nearest whole number.

Key Concepts

cylinder volume calculation

The volume of a cylinder tells us how much space is inside it. To find this, we use a simple formula: \[\begin{equation}V = \pi r^2 h\text{\end{equation}\], where \text{\(V\)} represents the volume, \text{\(r\)} is the radius of the cylinder’s base, \text{\(h\)} denotes the cylinder’s height, and \text{\pi\}otpi\text{ is approximately 3.1416.Let’s break down this formula step-by-step:

• Step 1: Find the radius, if not given directly (usually by dividing the diameter by two).
• Step 2: Square the radius ( \text{\(r^2\)}\text).
• Step 3: Multiply that squared result by the height ( \text{\(h\)}\text).
• Step 4: Multiply that result by \text{\(pi\)}\text.

Once you perform these steps, you'll get the cylinder's volume. Make sure to keep your units consistent!

radius of a cylinder

The radius is a key part of the cylinder volume calculation. For a cylinder, the radius ( \text{\(r\)}\text) is half the diameter of the base.
Here’s the conceptual breakdown:

• The diameter is the total distance across the circular base through the center.
• The radius is half of this distance.

For the Marina City example, the diameter is given as 35 feet. Thus, the radius \text{\(r = \frac{35}{2} = 17.5\)} feet. This value is crucial because it gets squared in the volume formula,\text{\( r^2\)}.\text>
Always double-check your radius calculation before moving on to ensure accuracy.

pi in geometry

Pi (\text{\(pi\)}\text) is an important constant in geometry representing the ratio of a circle’s circumference to its diameter. In numeric terms, \text{\(pi\)}≈ 3.1416.
This value is used whenever we are dealing with circles or their three-dimensional counterparts, like cylinders.
When calculating the volume of a cylinder, pi must be included to account for the circular base. For the volume formula \text{\( V = \pi r^2 h\)}, skipping pi or using an incorrect value would lead to a wrong answer.Remember, while pi is often approximated, using more decimal places as needed can yield a more precise result.

geometry problem solving

Solving geometry problems, like finding the volume of a cylinder, is a step-by-step process. Here’s a friendly guide to help you:

• Identify Given Values: Find any given dimensions, like the diameter and height.
• Calculate Necessary Measures: Convert all given values into the needed forms, such as calculating radius from the diameter.
• Apply Formulas: Insert your values into the geometry formulas. Ensure each variable is correctly placed.
• Perform Calculations: Work through the math step by step, paying attention to units and decimal places.
• Check and Round Results: After computing, verify the result’s accuracy. If necessary, round as specified (e.g., to the nearest whole number).

With these steps, you can approach any geometry problem with confidence. Just work through the details methodically and ensure each step is clear and correct. Happy problem-solving!