Problem 12 - Calculus: Early Transcendentals

Question

The base of a tetrahedron (a triangular pyramid) of height \(h\) is an equilateral triangle of side s. Its cross-sections perpendicular to an altitude are equilateral triangles. Express its volume \(V\) as an integral, and find a formula for \(V\) in terms of \(h\) and s. Verify that your answer is \((1 / 3)\) (area of base) \((\) height \() .\)

Step-by-Step Solution

Answer

The volume is \(\frac{\sqrt{3}}{12} s^2 h\), consistent with pyramid volume formula.

Step 1: Volume of a Pyramid

The volume of a pyramid can be calculated using the formula: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \] In this problem, the base is an equilateral triangle, and we need to express the volume in terms of an integral.

Step 2: Find Base Area

The area of an equilateral triangle with side length \(s\) is given by:\[ A = \frac{\sqrt{3}}{4} s^2 \] This will be the base area for our volume calculation.

Step 3: Relationship Between Height and Cross Sections

The tetrahedron has cross-sections that are equilateral triangles, perpendicular to its height \(h\). The side of these cross-sectional triangles changes as a linear function of their distance from the apex of the tetrahedron.

Step 4: Define Linear Function for Cross-Section Side

Let \(x\) be the distance from the vertex (apex) down to a particular cross-section. The side length of the equilateral triangle at any point \(x\) is \(\frac{s}{h}x\).

Step 5: Area of Cross-Section

The area of a cross-sectional equilateral triangle at distance \(x\) from the apex is given by:\[ A(x) = \frac{\sqrt{3}}{4} \left( \frac{s}{h}x \right)^2 = \frac{\sqrt{3}}{4} \frac{s^2}{h^2} x^2 \]

Step 6: Integrating Cross-Sectional Areas

The volume of the tetrahedron can be calculated as the integral of the cross-sectional areas from \(0\) to \(h\):\[ V = \int_0^h A(x) \, dx = \int_0^h \frac{\sqrt{3}}{4} \frac{s^2}{h^2} x^2 \, dx \]

Step 7: Integral Calculation

Calculate the integral:\[ V = \frac{\sqrt{3}}{4} \frac{s^2}{h^2} \int_0^h x^2 \, dx = \frac{\sqrt{3}}{4} \frac{s^2}{h^2} \left[ \frac{x^3}{3} \right]_0^h = \frac{\sqrt{3}}{4} \frac{s^2}{h^2} \times \frac{h^3}{3} \] \[ = \frac{\sqrt{3}}{12} s^2 h \]

Step 8: Verify Result

Check with the formula for the volume of a pyramid:\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} \times \frac{\sqrt{3}}{4} s^2 \times h \]\[ = \frac{\sqrt{3}}{12} s^2 h \] Both approaches give the same answer, verifying that they are consistent.

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics that helps us find the accumulation of quantities, such as areas under curves or volumes. When we talk about finding the volume of a shape using integrals, we are essentially summing up infinitesimally small pieces (in this case, cross-sectional areas) of the shape, from one point to another. This is particularly useful for objects with non-uniform shapes, such as a tetrahedron.

In this exercise, we use integral calculus to sum up the areas of cross-sectional equilateral triangles from the base to the apex of the tetrahedron. This approach involves integrating the area function, which varies with distance from the apex, over the entire height of the tetrahedron.

Equilateral Triangle

An equilateral triangle is a triangle with all three sides and angles being equal. This unique property makes calculating dimensions such as the area simpler, as we can use specific formulas tailored to its geometry.

For an equilateral triangle with side length \(s\), the area \(A\) is calculated using the formula: \( A = \frac{\sqrt{3}}{4} s^2 \). This is due to its symmetrical nature, allowing for straightforward geometric calculations. In the context of the tetrahedron, both the base and the perpendicular cross-sections are equilateral triangles, which simplifies the process of calculating the tetrahedron's volume.

Volume of a Pyramid

The volume of a pyramid, including tetrahedrons (triangular pyramids), can be calculated using the formula: \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \). This formula reflects the fact that a pyramid occupies one-third of the volume occupied by a prism with the same base area and height.

For a tetrahedron with an equilateral triangle base, once the area of the base is known, the formula simplifies the volume calculation. The triangular base's area is determined using the aforementioned formula for the equilateral triangle, and multiplying by the height and then dividing by three gives the volume of the tetrahedron.

Cross-Sectional Area

Cross-sectional areas refer to the areas of slices, or sections, of a solid object that are cut perpendicular to a specific axis. In the case of our tetrahedron problem, cross-sections are made perpendicular to the height.

As stated in the solution, these cross-sections are equilateral triangles whose side lengths decrease linearly from the base to the apex. The formula used for a cross-sectional equilateral triangle at a distance \(x\) from the apex is \( A(x) = \frac{\sqrt{3}}{4} \left( \frac{s}{h}x \right)^2 \). This formula demonstrates how the size of a cross-section changes as one moves vertically through the tetrahedron.

• The side length at distance \(x\) can be expressed as \(\frac{s}{h}x\).
• The cross-sectional area at each section is integrated from the base to the apex.

By accounting for each small equilateral triangle's area and summing them up (via integration), we find the total volume of the tetrahedron.