Question

What are the dimensions of the right circular cylinder with greatest curved surface area that can be inscribed in a sphere of radius \(r ?\)

Short answer

The dimensions are radius \( \frac{r}{\sqrt{2}} \) and height \( \sqrt{2}r \).

Step-by-step solution

Define Variables

Let the radius of the cylinder be \( x \) and the height be \( h \). The cylinder is inscribed in a sphere of radius \( r \), meaning that the cylinder fits perfectly within the sphere.

Establish Relationship Using Geometry

The diagonal of the cylinder (which is the diameter of the sphere) connects the top and bottom center of the cylinder over its lateral surface. By the Pythagorean theorem in the right triangle formed by the radius of the base, half of the height of the cylinder, and the radius of the sphere, we have: \[ x^2 + \left( \frac{h}{2} \right)^2 = r^2. \]

Express Height in Terms of Radius

Rearrange the equation from Step 2 to express the height \( h \) in terms of \( x \) and \( r \): \[ \frac{h}{2} = \sqrt{r^2 - x^2}, \] leading to \[ h = 2\sqrt{r^2 - x^2}. \]

Write the Function for Curved Surface Area

The formula for the curved surface area of a cylinder is \( 2\pi x h \). Substitute \( h = 2\sqrt{r^2 - x^2} \) from the previous step: \[ A(x) = 4\pi x \sqrt{r^2 - x^2}. \]

Differentiate and Find Critical Points

Differentiate \( A(x) \) with respect to \( x \): \[ A'(x) = \frac{d}{dx}[4\pi x (r^2 - x^2)^{1/2}] = 4\pi \left( \sqrt{r^2 - x^2} - \frac{x^2}{\sqrt{r^2 - x^2}} \right). \] Set \( A'(x) = 0 \) to find the critical points: \( \sqrt{r^2 - x^2} = \frac{x^2}{\sqrt{r^2 - x^2}} \).

Solve for Critical Points

From the equation \( r^2 - x^2 = x^2 \), solve for \( x \): \[ r^2 = 2x^2 \Rightarrow x^2 = \frac{r^2}{2} \Rightarrow x = \frac{r}{\sqrt{2}}. \] Substitute back to find \( h \): \[ h = 2\sqrt{r^2 - \left(\frac{r}{\sqrt{2}}\right)^2} = 2\sqrt{\frac{r^2}{2}} = \sqrt{2}r. \]

Verify Maximum Surface Area

By the nature of the function and its increase towards the critical value found, this is a maximum. Calculate second derivative if needed or verify graphically if critical point represents a maximum using calculus.

Key concepts

Inscribed Cylinder

When a cylinder is inscribed in a sphere, the cylinder fits perfectly inside the sphere without any part of it extending outside the sphere's surface. This creates a unique challenge of maximizing or optimizing certain measurements, like the curved surface area in the given problem. The inscribed cylinder's dimensions can be manipulated, but they must adhere to the sphere's radius constraint. To find these optimal dimensions, we start by defining the cylinder's radius and height. Considering the sphere's radius helps ensure that the entire cylinder fits within the boundaries of the sphere.

Curved Surface Area

The curved surface area of a cylinder is the area around the sides of the cylinder, leaving out the top and bottom circles. It's calculated using the formula: \(2\pi x h\), where \(x\) is the cylinder's radius and \(h\) is its height. In optimization problems, like ours, the aim is to maximize this area by finding the best possible values for \(x\) and \(h\). This usually involves expressing one variable in terms of the other, as derived from geometric constraints like being inscribed in a sphere. By substituting potential dimensions into this formula, we can calculate and compare the resulting areas.

Critical Points

In calculus, critical points are places where the derivative of a function is zero or undefined. These points often indicate where a function reaches a maximum or minimum value. To find the dimensions of the cylinder that give the greatest surface area, we need to calculate the derivative of the curved surface area function. Once critical points are found, it’s essential to determine whether they represent a maximum or minimum. This can be done by analyzing the second derivative or visually inspecting the function’s behavior around these points.

Differentiation

Differentiation is a fundamental tool in calculus used to find rates of change and solve optimization problems. By differentiating the surface area function with respect to the cylinder's radius \(x\), we obtain an expression that helps us find points where the surface area is neither increasing nor decreasing. These are the critical points. The differentiation process involves applying rules like the product rule, chain rule, or power rule, depending on the function's complexity. After obtaining the derivative, setting it to zero helps locate critical points, essential for optimizing the cylinder’s dimensions.

Pythagorean Theorem

The Pythagorean Theorem is used in this problem to relate the cylinder's dimensions to the sphere's radius. It states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Here, the hypotenuse is the sphere's radius, and the other two sides are half the cylinder's height and its base radius. Using the theorem, the relation \(x^2 + \left(\frac{h}{2}\right)^2 = r^2\) is established. This is essential for expressing the height \(h\) in terms of the radius \(x\) and finding optimal dimensions of the cylinder that fit inside the sphere.