Problem 32 - Elementary Algebra

Question

For Problems 11-32, use the geometric formulas given in this section to help solve the problems. (Objective 3 ) If the total surface area of a right circular cylinder is \(104 \pi\) square meters, and a radius of the base is 4 meters long, find the height of the cylinder.

Step-by-Step Solution

Answer

The height of the cylinder is 9 meters.

Step 1: Understand the Formula for Surface Area

The surface area \( S \) of a right circular cylinder can be given by the formula: \[ S = 2\pi r^2 + 2\pi rh \] where \( r \) is the radius of the base and \( h \) is the height of the cylinder.

Step 2: Substitute Known Values

We are given the total surface area \( S = 104\pi \) square meters and the radius \( r = 4 \) meters. Substitute these values into the surface area formula: \[ 104\pi = 2\pi (4)^2 + 2\pi (4)h \]

Step 3: Simplify the Equation

Simplify the given equation: \[ 104\pi = 2\pi \times 16 + 8\pi h \] which simplifies to \[ 104\pi = 32\pi + 8\pi h \].

Step 4: Solve for the Height \( h \)

Subtract \( 32\pi \) from both sides to isolate the term with \( h \): \[ 104\pi - 32\pi = 8\pi h \]. This simplifies to \[ 72\pi = 8\pi h \]. Divide both sides by \( 8\pi \) to solve for \( h \): \[ h = \frac{72\pi}{8\pi} = 9 \].

Step 5: Confirm the Solution

With \( h = 9 \), verify that substituting back into the original formula gives the correct total surface area: \[ S = 2\pi (4)^2 + 2\pi (4)(9) = 32\pi + 72\pi = 104\pi \], confirming the height is correct.

Key Concepts

Understanding Geometry of a Cylinder

In geometry, a cylinder is a 3D shape with two parallel circular bases connected by a curved surface. It resembles a can or a tube.
An essential feature of a cylinder is understanding its surface area, which consists of two parts:

• The circular top and bottom, also known as the bases.
• The curved surface that wraps around the sides, often called a lateral surface.

Visualizing these components helps in understanding how the total surface area is calculated. The formula for this is crucial since it combines the area of the two bases and the side.
Drawing or imagining the cylinder can make these components clearer and aid in grasping how geometry is applied to everyday objects.

Surface Area Formulas for Cylinders

Formulas are vital tools in geometry, allowing us to find properties like the surface area of shapes. For a cylinder, the surface area is determined using:\[ S = 2\pi r^2 + 2\pi rh \]Here,

• \(r\) represents the radius of the base.
• \(h\) stands for the height of the cylinder.
• \(2\pi r^2\) calculates the area of the two bases.
• \(2\pi rh\) gives the area of the lateral surface.

Understanding this formula means knowing it's combining the circles' and side's areas to find the total. Making sure to have the same units for measurements like radius and height is key to applying these formulas correctly.
This fundamental understanding of formulas is often necessary in solving more complex mathematics problems.

Problem Solving Using the Surface Area Formula

Problem-solving in mathematics often involves using well-known formulas as a base. In this case, we've been given the total surface area and need the cylinder's height. The "Step by Step" process shown provides a structure:

• First, substitute the known values into the formula for the surface area.
• Simplify the algebraic expression to isolate the desired variable, which is height \(h\) in this case.
• Solve the simplified equation to find \(h\).
• Check your solution by substituting back into the formula to ensure correctness.

Breaking down the problem in small, manageable steps makes it easier to tackle. You gradually work toward the solution, confirming your answer is correct.

The Role of Mathematics in Solving Geometric Problems

Mathematics acts as an essential framework for understanding and solving geometric challenges. It involves employing formulas, techniques, and logical reasoning.
When faced with finding parts of a shape, such as the height of a cylinder from its surface area, these skills are crucial:

• Recognition of known and unknown values is the starting point.
• Accurate substitution into formulas is critical to get the solution on track.
• Logical manipulation of equations allows isolating of desired outcomes.
• Verification ensures calculations were done accurately.

This entire mathematical process builds important analytic skills that are applicable beyond typical classroom problems. It's about understanding principles and applying them efficiently to real-world scenarios.