Problem 28 - Technical Mathematics with Calculus

Question

The frustum of a regular pyramid has square bases, one \(4.83 \mathrm{mm}\) on a side and the other \(2.84 \mathrm{mm}\) on a side. Its altitude is \(3.88 \mathrm{mm} .\) Find its volume and lateral area.

Step-by-Step Solution

Answer

The volume of the frustum is approximately 58.00 mm^3 and the lateral area is approximately 61.21 mm^2.

Step 1: Calculating the Volume of the Frustum

To find the volume of the frustum, use the formula: \[V = \frac{1}{3}h(a^2 + ab + b^2)\] where \(h\) is the altitude, \(a\) is the side of the larger base, and \(b\) is the side of the smaller base. Plug in the given values: \(h = 3.88\) mm, \(a = 4.83\) mm, \(b = 2.84\) mm, then calculate the volume.

Step 2: Calculating the Volume

Volume calculation: \[V = \frac{1}{3} \times 3.88 \times (4.83^2 + 4.83 \times 2.84 + 2.84^2)\] \[V = \frac{1}{3} \times 3.88 \times (23.3089 + 13.7142 + 8.0656)\] \[V = \frac{1}{3} \times 3.88 \times 45.0887\] \[V ≈ 58.00 \, \text{mm}^3\] (rounded to two decimal places)

Step 3: Calculating the Lateral Area of the Frustum

To calculate the lateral area, first find the slant height \(l\) using the Pythagorean theorem on the triangle formed by half the difference of the squares' sides, \((a-b)/2\), and the altitude \(h\). Slant height \(l = \sqrt{h^2 + \left(\frac{a - b}{2}\right)^2}\). Use this \(l\) and the perimeters of the bases to find the lateral area: \[A_L = \frac{P_a + P_b}{2} \times l\] where \(P_a\) and \(P_b\) are the perimeters of the larger and smaller bases respectively.

Step 4: Calculating the Slant Height

Calculate the slant height: \[l = \sqrt{3.88^2 + \left(\frac{4.83 - 2.84}{2}\right)^2}\] \[l = \sqrt{15.0544 + \left(0.995\right)^2}\] \[l = \sqrt{15.0544 + 0.990025}\] \[l ≈ 3.99 \, \text{mm}\] (rounded to two decimal places)

Step 5: Calculating the Lateral Area

Perimeters are \(P_a = 4 \times a\) and \(P_b = 4 \times b\). Using the slant height, calculate lateral area: \[A_L = \frac{4 \times 4.83 + 4 \times 2.84}{2} \times 3.99\] \[A_L = \frac{19.32 + 11.36}{2} \times 3.99\] \[A_L = 15.34 \times 3.99\] \[A_L ≈ 61.21 \, \text{mm}^2\] (rounded to two decimal places)

Key Concepts

Volume of a Frustum Formula

Understanding the volume of a frustum is a fundamental concept in geometry. A frustum is a portion of a solid—typically a cone or pyramid—that remains after cutting the top off with a plane parallel to the base.

For the volume of a frustum derived from a pyramid with a square base, the formula is: \[\begin{equation}V = \frac{1}{3} h (a^2 + ab + b^2)\end{equation}\]where:

• \(V\) is the volume of the frustum,
• \(h\) is the altitude or height of the frustum,
• \(a\) and \(b\) are the lengths of the sides of the larger and smaller square bases, respectively.

To apply this formula effectively, you need to measure or be provided with the dimensions of both bases and the perpendicular distance (altitude) between them. Calculating the volume then becomes a straightforward substitution of these values into the formula.

Calculating Lateral Area

The lateral area of a frustum is another intriguing concept to explore. It refers to the total area of the sides of the frustum, excluding its top and bottom bases. Calculating the lateral area can be approached by finding the slant height and using the perimeters of both bases.

To compute the lateral area (\(A_L\)) of a frustum with square bases, the formula used is:\[\begin{equation}A_L = \frac{P_a + P_b}{2} \times l\end{equation}\]where:

• \(P_a\) and \(P_b\) are the perimeters of the larger and smaller bases, and
• \(l\) is the slant height of the frustum.

This method effectively wraps a diagonal 'belt' around the middle of the frustum and multiplies it by the slant height to find the total lateral surface area. Remember to calculate both perimeters and slant height before finding the area.

Pythagorean Theorem

The Pythagorean Theorem is a timeless principle in geometry, vital for solving problems involving right triangles. This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In the mathematical form, the theorem is:\[\begin{equation}c^2 = a^2 + b^2\end{equation}\]where:

• \(c\) is the hypotenuse,
• \(a\) and \(b\) are the other two sides of the triangle.

In the context of a frustum, this theorem helps us find the slant height, which is essentially the hypotenuse of a right triangle formed by the altitude and half of the difference between the sides of the two bases.

Slant Height of a Frustum

Slant height is a necessary measurement when working with frustums as it's instrumental in calculating the lateral area. The slant height (\(l\)) of a frustum is the shortest distance from any point on the edge of one base to the edge of the other base, measured along the surface of the frustum.

Using the Pythagorean Theorem, the slant height can be found with the formula:\[\begin{equation}l = \sqrt{h^2 + \left(\frac{a - b}{2}\right)^2}\end{equation}\]where:

• \(h\) is the altitude and
• \(a\) and \(b\) are the sides of the larger and smaller bases, respectively.

Computing the slant height requires calculating the distance on the plane between the centers of the two bases, then utilizing this value and the altitude to form a right triangle and apply the Pythagorean Theorem. The slant height plays a key role in the overall geometry of a frustum, making it a crucial step in finding both volume and lateral area.