Problem 99 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Volume of a Cylinder The volume of a right circular cylinder is jointly proportional to the square of the radius of the circular base and to the height. If the volume is 300 cubic centimeters when the height is 10.62 centimeters and the radius is 3 centimeters, approximate the volume of a cylinder with radius 4 centimeters and height 15.92 centimeters. (image can't copy)

Step-by-Step Solution

Answer

The volume of the new cylinder is approximately 716.2 cm³.

Step 1: Identify the Formula

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

Step 2: Establish the Given Values

From the problem statement, we know that when \( r = 3 \) cm and \( h = 10.62 \) cm, \( V = 300 \) cubic centimeters.

Step 3: Calculate the Constant of Proportionality

Plug the given values into the formula to find the constant of proportionality, \( k \):\[ 300 = k \pi (3)^2 (10.62) \]Simplify and solve for \( k \):\( 300 = 9\pi \times 10.62 \times k \) which gives \( k = \frac{300}{9\times 10.62\pi} \approx 0.897 \).

Step 4: Apply the Constant to the New Values

Now use the calculated constant \( k \) to find the new volume \( V' \) with \( r = 4 \) cm and \( h = 15.92 \) cm:\[ V' = \pi (4)^2 (15.92) \times k \]\( V' = 16 \pi \times 15.92 \times 0.897 \).

Step 5: Solve for the New Volume

Calculate the new volume:\[ V' = 16 \times 15.92 \times 0.897 \times \pi \]This results in the approximation \( V' \approx 716.2 \text{ cm}^3 \) when rounded to one decimal place.

Key Concepts

Constant of Proportionality

The constant of proportionality is a crucial factor that links quantities in mathematical relationships. In the case of the volume of a cylinder, the volume is jointly proportional to the square of the radius and the height. This means that there is a fixed ratio, or constant (\(k\)), that relates these variables.

Finding this constant involves setting up an equation using known values and solving for \(k\). For example, when you have a volume (\(V = 300 \text{cm}^3\)), a radius (\(r = 3 \text{cm}\)), and a height (\(h = 10.62 \text{cm}\)), you plug these into the formula \(V = k \pi r^2 h\).

By solving \(300 = k\pi (3)^2(10.62)\), you isolate \(k\) to find it is approximately 0.897 as the constant of proportionality. This is now a fixed number you can use again to find unknown volumes, given similar relationships of radius and height.

Joint Variation

Joint variation describes a scenario where one variable is directly proportional to the multiplication of two or more other variables. In the context of the volume of a cylinder, this concept is demonstrated because the volume varies jointly with the square of the radius of the base and the height.

Mathematically, this is expressed as \(V = k\pi r^2h\), indicating that any change in either the radius or the height will proportionally affect the volume. If you increase the radius or the height, the volume increases, and vice versa.

• When the radius doubles, the volume increases by a factor of four, as the radius is squared in the equation.
• If the height is tripled, the volume triples accordingly.

Understanding joint variation helps us make accurate predictions about the effects of changing dimensions on the volume.

Geometric Formulas

Geometric formulas are essential mathematical tools used to calculate various dimensions and volumes, such as the volume of a cylinder. The formula for a cylinder's volume is expressed as \(V = \pi r^2h\), where \(r\) is the radius and \(h\) is the height. This formula is derived from geometrical principles that consider the area of the circular base (\(\pi r^2\)) and how that area extends throughout the height of the cylinder.

To effectively use geometric formulas, you must first understand the shape and its relevant dimensions. For cylinders:

• The base is a circle, so the primary part of the formula involves calculating the area of this circle (\(\pi r^2\)).
• The height (\(h\)) extends this base area upwards, determining the overall size of the cylinder.

By applying these concepts, you can find volumes or other measurements for simple and complex shapes. Practicing with these formulas strengthens your ability to solve geometrical problems effectively.