Finding Cylinder Height and Radius Relationship from Surface Areas

This problem asks us to find the relationship between the height (h) and radius (r) of a solid right circular cylinder based on a given condition involving its surface areas. The condition is that two times the total surface area is three times its curved surface area.

To solve this, we need to recall the formulas for the surface areas of a right circular cylinder:

• The curved surface area (CSA) is the area of the side wall, which is given by \( \text{CSA} = 2\pi rh \), where \(r\) is the radius of the base and \(h\) is the height.
• The total surface area (TSA) is the sum of the curved surface area and the areas of the two circular bases. The area of each base is \( \pi r^2 \). So, the total surface area is \( \text{TSA} = 2\pi rh + 2\pi r^2 \).

The problem states that two times the total surface area is equal to three times the curved surface area. We can write this as an equation:

\( 2 \times (\text{TSA}) = 3 \times (\text{CSA}) \)

Now, substitute the formulas for TSA and CSA into this equation:

\( 2 \times (2\pi rh + 2\pi r^2) = 3 \times (2\pi rh) \)

Let's simplify and solve this equation step-by-step to find the relationship between \(h\) and \(r\):

1. Expand both sides of the equation:

   \( 4\pi rh + 4\pi r^2 = 6\pi rh \)

2. We want to isolate the terms involving \(h\) and \(r\). Notice that \( \pi \) and \( r \) are common factors in all terms (assuming \(r \ne 0\), which must be true for a cylinder). We can divide both sides by \( \pi r \).

   \( \frac{4\pi rh + 4\pi r^2}{\pi r} = \frac{6\pi rh}{\pi r} \)

   This simplifies to:

   \( 4h + 4r = 6h \)

3. Now, gather the terms involving \(h\) on one side and the terms involving \(r\) on the other side. Subtract \(4h\) from both sides:

   \( 4r = 6h - 4h \)

   \( 4r = 2h \)

4. To find the relationship between \(h\) and \(r\), we can divide both sides by 2:

   \( \frac{4r}{2} = \frac{2h}{2} \)

   \( 2r = h \)

   Or equivalently,

   \( h = 2r \)

So, the height \(h\) of the cylinder is equal to two times its radius \(r\).

Let's check the given options:

• Option 1: \( h = r \) (Incorrect)
• Option 2: \( h = 2r \) (Correct)
• Option 3: \( 2h = 3r \) (Incorrect)
• Option 4: \( 3h = 4r \) (Incorrect)

The relationship derived from the given condition is \( h = 2r \).

Understanding the formulas for surface area and being able to set up and solve equations based on given conditions are key skills for solving geometry problems involving solid shapes like cylinders.

Revision Table: Cylinder Surface Area Formulas

This table summarizes the essential formulas used for solving problems related to the surface area of a solid right circular cylinder.

Additional Information: Properties of Right Circular Cylinders

A solid right circular cylinder is a three-dimensional shape with two parallel circular bases of the same size and a curved surface connecting them. The axis joining the centers of the two bases is perpendicular to the bases, which is why it's called a "right" cylinder. Understanding these basic properties helps in visualizing the shape and applying the correct formulas.

• The radius \(r\) is the radius of the circular base.
• The height \(h\) is the perpendicular distance between the two bases.
• The volume of a cylinder is given by \( V = \pi r^2 h \).
• The curved surface, if unrolled, forms a rectangle with width equal to the height \(h\) and length equal to the circumference of the base \( 2\pi r \). The area of this rectangle is \( (2\pi r) \times h = 2\pi rh \), which confirms the CSA formula.

Problems involving cylinders often require manipulating these formulas based on given conditions, similar to what was done in this problem involving the relationship between TSA and CSA.