Problem 63 - Precalculus : Building Concepts and Connections

Question

You will use polynomials to study real-world problems. Manufacturing The height of a right circular cylinder is 5 inches more than its radius. Find the dimensions of the cylinder if its volume is 1000 cubic inches.

Step-by-Step Solution

Answer

After carrying out the above steps, we find the dimensions (radius r and height h) of the cylinder. However, the numerical solution depends on carrying out the calculations from step 3.

Step 1: Translate the problem into an equation

First, translate the prompt into a mathematical equation using the volume formula \( V = \pi \times r^2 \times h \). Substitute 1000 for the volume V, and for the height, use \( r+5 \) (since the height is 5 inches more than the radius), making the equation \( 1000 = \pi \times r^2 \times (r+5) \).

Step 2: Solve for r

Next, solving this equation for \( r \) involves dividing both sides of the equation by pi: \( \frac{1000}{\pi} = r^2 \times (r+5) \). This equation is trinomial, and it needs to be solved either by factoring or using the quadratic formula. If the equation is difficult to factor, use the quadratic formula.

Step 3: Use the quadratic formula to solve for r

The quadratic formula is \( \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Remember that a, b and c are the coefficients in a trinomial equation. Rewriting the equation in the form \( ar^2 + br + c = 0 \), you will have \( r^2 + 5r - \frac{1000}{\pi} \). Therefore, \( a = 1, b = 5, and c = - \frac{1000}{\pi} \). Now substitute these values into the quadratic formula and solve for r which represents the radius. Discard the negative root since negative numbers would not make sense in this context.

Step 4: Calculate height (h)

Finally, using the positive root, add 5 to find the value of h (height of the cylinder). Thus, the radius and height of the cylinder have been found.

Key Concepts

Volume of Cylinder

Understanding the volume of a cylinder is essential in solving real-world problems involving cylindrical shapes. A right circular cylinder's volume is calculated using the formula: \[ V = \pi \times r^2 \times h \] Where:

• \( V \) is the volume.
• \( r \) is the radius of the base.
• \( h \) is the height of the cylinder.

This formula multiplies the area of the circular base, \( \pi \times r^2 \), by the height, \( h \), to find the total space inside the cylinder. For example, if you have a cylinder with a radius of 3 inches and a height of 7 inches, you can find the volume by plugging into the formula. Calculating this helps solve problems where you need to know how much space is inside a cylindrical object, like a can or a pipe.

Quadratic Formula

The quadratic formula is a powerful tool for solving equations where factoring is difficult. It can solve any quadratic equation of the form:\[ ax^2 + bx + c = 0 \]Here's the formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Where:

• \( a, b, \) and \( c \) are coefficients from the equation.
• \( x \) represents the unknowns or the solutions.

The '±' symbol means there will usually be two solutions. One solution uses the plus sign, and the other uses the minus sign. When using the quadratic formula:

• First, find \( b^2 - 4ac \), known as the discriminant.
• If the discriminant is positive, there are two real solutions.
• If it's zero, there is one real solution.
• If negative, the solutions are not real numbers.

The quadratic formula always works, which makes it a reliable method for solving quadratic equations.

Trinomial Equation

A trinomial equation is a type of polynomial with three terms. They look like this:\[ ar^2 + br + c = 0 \]In these equations:

• \( a \), \( b \), and \( c \) are constants.
• \( r \) is the variable you want to solve for.

These polynomials can often be factored into simpler expressions, but when they can't, the quadratic formula is used. In the context of solving for the radius of a cylinder, you solve a trinomial equation formed by plugging in the known volume and relating height to the radius (like adding 5). This transforms the problem into a quadratic one, needing these methods to find the correct values for the dimensions.

Real-world Applications

Polynomials, especially quadratic equations, have numerous real-world applications. For instance, determining the dimensions of a cylinder from its volume might be encountered in fields like manufacturing, engineering, or design. Some practical use-cases include:

• Manufacturing: Planning the size of cans or components.
• Architecture: Calculating storage tank capacities.
• Engineering: Designing mechanical parts with specific volume requirements.

Understanding how to translate real-life conditions into mathematical equations helps in creating solutions and designs that meet specified needs efficiently. It shows the power of mathematics in everyday technology and design challenges.