Problem 39 - Thomas Calculus

Question

A cone problem Begin with a circular piece of paper with a 4 in. radius as shown in part (a). Cut out a sector with an arc length of \(x\) . Join the two edges of the remaining portion to form a cone with radius \(r\) and height \(h,\) as shown in part (b). Graph cannot copy a. Explain why the circumference of the base of the cone is \(8 \pi-x .\) b. Express the radius \(r\) as a function of \(x\) . c. Express the height \(h\) as a function of \(x\) . d. Express the volume \(V\) of the cone as a function of \(x\) .

Step-by-Step Solution

Answer

The volume of the cone as a function of \(x\) is \(V(x) = \frac{1}{3}\pi\left(4 - \frac{x}{2\pi}\right)^2\sqrt{16 - \left(4 - \frac{x}{2\pi}\right)^2}.\)

Step 1: Understand the Geometry

A circular piece of paper with a radius of 4 inches is used to form the cone. When a sector with arc length \(x\) is cut out, the remaining part is the base to be shaped into a cone.

Step 2: Calculate the Remaining Circumference

The total circumference of the initial circle is \(2 \pi \times 4 = 8\pi\). Removing a sector with arc length \(x\) leaves the remaining arc length as the circumference of the base of the cone: \(8\pi - x\).

Step 3: Express Radius as a Function of x

The circumference of the base of the cone is given by \(2\pi r = 8\pi - x\). Solving for \(r\), we have: \[ r = \frac{8\pi - x}{2\pi} = 4 - \frac{x}{2\pi}. \] Thus, \(r\) as a function of \(x\) is \(r(x) = 4 - \frac{x}{2\pi}\).

Step 4: Use Pythagorean Theorem to Find Height

The slant height of the cone is the radius of the original circle (4 inches). Using the Pythagorean theorem, \(r^2 + h^2 = 4^2\). We express \(h\) as: \( h = \sqrt{4^2 - r^2} = \sqrt{16 - \left(4 - \frac{x}{2\pi}\right)^2}. \)

Step 5: Substitute Radius into Height Formula

Substitute \(r(x) = 4 - \frac{x}{2\pi}\) into the height equation: \[ h(x) = \sqrt{16 - \left(4 - \frac{x}{2\pi}\right)^2}. \] Simplifying further if needed provides \(h\) as a function of \(x\).

Step 6: Express Volume as a Function of x

The volume of a cone \(V\) is given by \(V = \frac{1}{3}\pi r^2h\). Using \(r(x)\) and \(h(x)\), substitute to express \(V\) as a function of \(x\): \[ V(x) = \frac{1}{3} \pi \left(4 - \frac{x}{2\pi}\right)^2 \sqrt{16 - \left(4 - \frac{x}{2\pi}\right)^2}.\]

Key Concepts

Geometry

Geometry is a fascinating aspect of mathematics that deals with shapes and sizes. When considering objects like a cone, understanding its geometric structure becomes important. In our exercise, we started with a circular piece of paper and transformed it into a cone by cutting out a sector. The geometric challenge lies in understanding how the reduction (cutting a sector) affects other parts like the base and height of the cone.
Understanding how geometric properties relate to each other helps solve complex problems. For example, recognizing that cutting an arc affects the circumference is an essential geometric insight. Applying properties like circumference and recognizing that the remaining arc forms the base is a classic geometry application.
Thus, geometry helps visualize transformations and relationships between different aspects of shapes.

Cone Volume Calculation

Calculating the volume of a cone involves understanding the formula and the specifics of the shape in question. For a cone, the volume formula is \[V = \frac{1}{3} \pi r^2 h\]Here, \(r\) represents the radius of the base, and \(h\) is the height.
To apply this formula, you need both the radius and the height as inputs. This typically involves expressing these dimensions in terms of another variable, such as \(x\) in our problem.
The key step is substituting the expressions for \(r\) and \(h\) in terms of \(x\) from previous calculations into this formula. This ultimately gives the volume of the cone as a function of \(x\), showcasing how each alteration or transformation in shape affects the volume.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates to right-angled triangles. For any right triangle, it states:
\[a^2 + b^2 = c^2\]where \(c\) is the hypotenuse (longest side), and \(a\) and \(b\) are the other two sides.
In our exercise of forming a cone, the slant height (which is the radius of the original circle) is particularly crucial. This serves as the hypotenuse in a right triangle formed by the height and the radius of the base.
Using the Pythagorean Theorem allows us to calculate the height \(h\) from the radius \(r\), ensuring we have all necessary measures to determine the cone's properties such as volume. It's a straightforward application that links algebraic expressions to geometric intuition, making it easier to solve the problem.

Functions of a Variable

Functions of a variable involve expressing one quantity in terms of another. This is pivotal in complex mathematical problems where direct measurements are not feasible.
In our exercise, this concept allows us to express both the radius \(r\) and the height \(h\) in terms of a single variable \(x\). This simplification transforms our understanding and ability to calculate attributes like the cone's volume efficiently.
By defining functions \(r(x)\) and \(h(x)\), we essentially adjust the familiar geometric formulas to be adaptable with the variable \(x\). This functional approach underpins many areas of calculus and enables dynamic analysis of shapes based on their transformations.