Problem 63 - College Algebra with Corequisite Support

Question

For the following exercises, determine the function described and then use it to answer the question. The surface area, \(A,\) of a cylinder in terms of its radius, \(r,\) and height, \(h,\) is given by \(A=2 \pi r^{2}+2 \pi r h .\) If the height of the cylinder is 4 feet, express the radius as a function of \(A\) and find the radius if the surface area is 200 square feet.

Step-by-Step Solution

Answer

The radius of the cylinder is approximately 1.184 feet.

Step 1: Substitute the Height

Given that the height of the cylinder is 4 feet, substitute 4 for the height, \(h\), in the surface area equation:\[A = 2\pi r^2 + 2\pi r \times 4\] Which simplifies to:\[A = 2\pi r^2 + 8\pi r\]

Step 2: Rearrange to Solve for Radius

The equation \(A = 2\pi r^2 + 8\pi r\) is in terms of \(A\) and \(r\). Rearrange it to express \(r\) in terms of \(A\).Rearrange to get a standard quadratic form:\[2\pi r^2 + 8\pi r - A = 0\].This is a quadratic equation in \(r\).

Step 3: Use Quadratic Formula

To solve for \(r\), use the quadratic formula:\[r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \(a = 2\pi\), \(b = 8\pi\), and \(c = -A\):\[r = \frac{-8\pi \pm \sqrt{(8\pi)^2 - 4 \cdot 2\pi \cdot (-A)}}{2 \cdot 2\pi}\].

Step 4: Simplify the Expression

Substitute \(A = 200\) into the quadratic formula:\[r = \frac{-8\pi \pm \sqrt{(8\pi)^2 - 8\pi \cdot (-200)}}{4\pi}\].Calculate under the square root:\[r = \frac{-8\pi \pm \sqrt{64\pi^2 + 1600\pi}}{4\pi}\].

Step 5: Calculate the Discriminant

Simplify further by evaluating the discriminant:\[64\pi^2 + 1600\pi\].Compute it:\[64\pi^2 + 1600\pi\approx 1601.74\], assuming \(\pi \approx 3.14\).

Step 6: Solve for the Radius

Finally, perform the calculations:\[r = \frac{-8\pi \pm \sqrt{1601.74}}{4\pi}\].Since radius cannot be negative, only consider the positive solution:\[r \approx \frac{-25.12 + 40.02}{12.56}\] which is approximately 1.184 feet.

Key Concepts

Quadratic Equation

A quadratic equation is an algebraic expression of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable.

Quadratic equations have a degree of 2, which means the highest exponent of the unknown is 2. This form makes it quite common in many mathematical problems and applications.

There are several ways to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. The quadratic formula is particularly handy when the equation does not factor neatly.

Radius as a Function of Surface Area

In geometry, cylinders have both a curved lateral surface and two circular bases. The surface area \(A\) of a cylinder can be expressed in terms of its radius \(r\) and height \(h\).

For a given height, the surface area formula becomes: \(A = 2\pi r^2 + 2\pi rh\).

To express the radius as a function of surface area, we rearrange this equation to solve for \(r\). By treating the formula as a quadratic in \(r\), it becomes easier to manipulate.

This is crucial for problems where the surface area is known, and we need to find the cylinder's radius. Understanding this relationship allows us to apply the quadratic formula effectively.

Cylinder Geometry

Cylinders are three-dimensional shapes with two parallel circular bases connected by a curved surface at a fixed distance apart. They possess symmetrical properties and are an essential concept in geometry.

The main components of a cylinder are:

• Height \(h\) - the distance between the two bases.
• Radius \(r\) - the radius of the circular base.
• Surface Area \(A\) - combining the lateral area and the areas of the two bases.
• Volume \(V\) - calculated by \(\pi r^2 h\).

Understanding these properties aids in solving geometric problems involving cylinders, from computing surface area to determining dimensions given certain conditions.

Quadratic Formula

The quadratic formula provides a reliable way to solve any quadratic equation. It is given as: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation \(ax^2 + bx + c = 0\).

The formula incorporates the discriminant \(b^2 - 4ac\), which determines the nature of the roots:

• If the discriminant is positive, the equation has two real and distinct roots.
• If zero, there is exactly one real root.
• If negative, the roots are complex and not real.

It's a universal solution approach and especially useful when other methods, like factoring, are not applicable.

By substituting the values from the cylinder's surface area problem, we calculated the radius \(r\) efficiently using this formula, ensuring the solution was both accurate and straightforward.