Question

Develop a formula for radius as a function of surface area for a) a cylinder with equal diameter and height b) a cone with height three times its diameter

Short answer

Cylinder: \(r = \sqrt{\frac{A_c}{6\pi}}\). Cone: \(r = \sqrt{\frac{A_k}{\pi (1 + \sqrt{37})}}\).

Step-by-step solution

Identify the given relationships

For the cylinder, the diameter and height are equal. Hence, if the radius is denoted by \(r\), then the diameter \(d\) is \(2r\) and the height \(h\) is also \(2r\). For the cone, the height is three times the diameter. Thus, if the radius is \(r\), then the diameter \(d\) is \(2r\) and the height \(h\) is \(6r\).

Write the formula for the surface area of the cylinder

The surface area \(A_c\) of a cylinder is given by: \[ A_c = 2\pi r h + 2\pi r^2 \] Substitute \(h = 2r\) into the equation: \[ A_c = 2\pi r (2r) + 2\pi r^2 = 4\pi r^2 + 2\pi r^2 = 6\pi r^2 \]

Solve for the radius of the cylinder

To find the radius \(r\) as a function of surface area \(A_c\): \[ A_c = 6\pi r^2 \] Divide both sides by \(6\pi\): \[ r^2 = \frac{A_c}{6\pi} \] Take the square root of both sides: \[ r = \sqrt{\frac{A_c}{6\pi}} \]

Write the formula for the surface area of the cone

The surface area \(A_k\) of a cone is given by: \[ A_k = \pi r^2 + \pi r \, l \] where \(l\) is the slant height. The slant height \(l\) can be found using the Pythagorean theorem: \[ l = \sqrt{r^2 + h^2} \] Substitute \(h = 6r\) into the equation: \[ l = \sqrt{r^2 + (6r)^2} = \sqrt{r^2 + 36r^2} = \sqrt{37r^2} = r\sqrt{37} \] Thus, \[ A_k = \pi r^2 + \pi r (r\sqrt{37}) = \pi r^2 + \pi r^2 \sqrt{37} = \pi r^2 (1 + \sqrt{37}) \]

Solve for the radius of the cone

To find the radius \(r\) as a function of surface area \(A_k\): \[ A_k = \pi r^2 (1 + \sqrt{37}) \] Divide both sides by \(\pi (1 + \sqrt{37})\): \[ r^2 = \frac{A_k}{\pi (1 + \sqrt{37})} \] Take the square root of both sides: \[ r = \sqrt{\frac{A_k}{\pi (1 + \sqrt{37})}} \]

Key concepts

cylinder surface area

The surface area of a cylinder comprises two main parts: the curved surface area and the area of the two circular bases. A cylinder's dimensions are specified by its radius (r) and height (h). The formula for the cylinder's surface area is given by:

\[ A_c = 2\pi r h + 2\pi r^2 \]
The first part of the equation, 2πrh, calculates the curved surface area, while the second part, 2πr², accounts for the top and bottom circular bases.

To illustrate with an example, if the diameter of a cylinder is the same as its height, the height can be represented as 2r. Substituting this value into the formula, we get: \[ A_c = 2\pi r (2r) + 2\pi r^2 = 6\pi r^2 \]
This formula is crucial for calculating the total surface area based on the given radius.

cone surface area

The surface area of a cone includes the base area and the lateral (curved) surface area. A cone's dimensions are defined by its radius (r) and height (h). The formula for the total surface area of a cone is: \[ A_k = \pi r^2 + \pi r l \]
Here, πr² represents the base area, and πrl represents the lateral surface area. The term 'l' is the slant height, which needs to be calculated separately.

Given a cone with a height that is three times its diameter, which can be expressed as h = 6r, we find the slant height (l) using the Pythagorean theorem: \[ l = \sqrt{r^2 + h^2} = \sqrt{r^2 + (6r)^2} = \sqrt{37r^2} = r \sqrt{37} \]
Thus, the surface area formula becomes: \[ A_k = \pi r^2 + \pi r (r \sqrt{37}) = \pi r^2 (1 + \sqrt{37}) \]
This formula is essential for finding the cone's surface area based on the given radius.

radius calculation

To solve for the radius based on the surface area, we use the surface area formulas and manipulate them to isolate 'r'. Let's start with the cylinder:

Given the surface area formula \[ A_c = 6\pi r^2 \], we solve for r: \[ r^2 = \frac{A_c}{6\pi} \] Taking the square root of both sides: \[ r = \sqrt{\frac{A_c}{6\pi}} \]
For the cone, given the surface area formula \[ A_k = \pi r^2 (1 + \sqrt{37}) \], we isolate r: \[ r^2 = \frac{A_k}{\pi (1 + \sqrt{37})} \] Taking the square root of both sides: \[ r = \sqrt{\frac{A_k}{\pi (1 + \sqrt{37})}} \]
These formulas help calculate the radius when the surface area of either shape is known, making them critical in geometry and various practical applications.

slant height

The slant height of a shape, particularly in a cone, is an important dimension that often needs to be calculated using the Pythagorean theorem. For a cone, the slant height 'l' is the hypotenuse of a right triangle formed by the radius (r) and the height (h) of the cone.

The Pythagorean theorem states that \ l = \sqrt{r^2 + h^2} \ . Given a cone with height h = 6r, we substitute this into the formula: \[ l = \sqrt{r^2 + (6r)^2} = \sqrt{r^2 + 36r^2} = \sqrt{37r^2} = r \sqrt{37} \]
Understanding and calculating the slant height is crucial as it directly affects the calculation of the cone's lateral surface area and, subsequently, the total surface area.