Problem 10 - Thomas Calculus

Question

Find the lateral surface area of the cone generated by revolving the line segment \(y = x / 2,0 \leq x \leq 4 ,\) about the \(y\) -axis. Check your answer with the geometry formula Lateral surface area \(= \frac { 1 } { 2 } \times\) base circumference \(\times\) slant height.

Step-by-Step Solution

Answer

The lateral surface area is \(4\pi\sqrt{5}\).

Step 1: Define the Elements of the Cone

The line segment given is \(y = \frac{x}{2}\) from \(x = 0\) to \(x = 4\). By revolving this around the \(y\)-axis, a cone is formed. The end of the line at \(x = 4\) gives the height of the cone as \(y = \frac{4}{2} = 2\). This is the radius of the base of the cone. The slant height can be calculated using the Pythagorean theorem based on this radius and the height along the \(x\) direction, which is 4.

Step 2: Compute the Slant Height

The slant height \(l\) of the cone is the hypotenuse of the right triangle formed by the radius and the \(x\) component of the line. Using the Pythagorean theorem: \(l = \sqrt{r^2 + h^2}\), where the base radius \(r = 2\) and height \(h = 4\). Thus, \(l = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}\).

Step 3: Calculate the Lateral Surface Area Using Geometry Formula

The formula for lateral surface area of a cone is \( \frac{1}{2} \times 2\pi r \times l\). Substituting the known values, we get: \( \frac{1}{2} \times 2\pi \times 2 \times 2\sqrt{5} = 2\pi \cdot 2 \sqrt{5} = 4\pi\sqrt{5}\). Hence, the lateral surface area is \(4\pi\sqrt{5}\).

Step 4: Validate the Calculation

Cross-checking with the geometry formula confirms the lateral surface area previously computed in Step 3: \(4\pi\sqrt{5}\). This validation ensures the accuracy of the computation.

Key Concepts

Cone

A cone is a three-dimensional geometric shape that has a circular base and a pointed top called the apex. Imagine an ice cream cone, where the wide end is the base that sits on your hand, and the pointed end is where the ice cream is! Cones have three main components:

• Base: The flat, circular part of the cone.
• Height: The perpendicular distance from the base to the apex.
• Slant Height: The diagonal distance from any point on the base to the apex.

Cones can be classified as right or oblique, but when we deal with classic geometry problems, we often are focusing on right circular cones. In this problem, when a line is revolved around an axis, it generates a cone, reflecting real-world scenarios like ice cream cones or party hats.

Revolution of a Line Segment

The revolution of a line segment is a fascinating geometrical concept. It involves rotating a straight line around a fixed axis, creating a three-dimensional shape. In our problem, the line segment is given by the equation \(y = \frac{x}{2}\) and is revolved around the \(y\)-axis.
This revolution produces a cone. The line segment determines the dimensions of the cone:

• The base radius is obtained from the line's endpoint at \(x = 4\), giving us \(y = 2\).
• The height refers to the original straight line's length along the x-axis, from 0 to 4 in our case.

By understanding how the line rotates around the axis, one can visualize the cone's structure, which is key to solving tasks related to finding surface areas or volumes.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry. It relates the sides of a right triangle where \(a^2 + b^2 = c^2\), with \(c\) being the hypotenuse.
In the context of cones, this theorem helps determine the slant height. For our cone problem, we identify the right triangle comprising:

• One side as the radius \(r = 2\).
• Another side as the height \(h = 4\).
• The slant height serves as the hypotenuse \(l\).

Using these values, we apply the Pythagorean theorem to find \(l = \sqrt{2^2 + 4^2} = \sqrt{20} = 2\sqrt{5}\). This calculation is crucial for determining the lateral surface area of the cone.

Slant Height

The slant height is an essential measurement in a cone. It is the diagonal length from any point on the base circle to the apex. In 3D geometry, it is analogous to the hypotenuse of a triangle formed by the height and radius of the cone.
For our cone formed by rotating the line \(y = \frac{x}{2}\), the slant height was found using the Pythagorean theorem and discovered as \(2\sqrt{5}\). This length is critical when calculating the lateral surface area.

• It connects the radius \(r\) and height \(h\) of the cone.
• It influences various formulas, like the one for lateral surface area: \(\frac{1}{2} \times\) base circumference \(\times\) slant height.

Knowing the slant height allows us to have a complete understanding of the cone's dimensions, enabling us to compute precise measurements of its surface.