Wemazi

Topic 8.8

Identify the dimensions of the slice

Builds on building the area of a square or rectangle slice. 6 minutes.

One base, three faces

A solid stands on the region between the x-axis and the parabola \(y = 9 - x^2\). A cross section, or slice, is a flat face that stands on the floor at one value of \(x\). Its base is the orange segment on the floor, 8 long at \(x = 1\).

The upright orange segment rises from the middle of the base to the highest point of the face. The first face is a semicircle, so the base is its diameter and the upright segment is its radius, half of 8, which is 4. Drag the slice, and the radius stays half of the base.

Press the second button. The face is an equilateral triangle, with three equal sides, so the base is a side, 8. The upright segment is the height, \(4\sqrt{3}\), less than the side.

Press the third button. The face is a right triangle with two equal legs, and its longest side, the hypotenuse, is the base. The height is half of the hypotenuse, 4.

One length on the floor

The orange segment \(w\) on the floor is the top boundary of the region minus the bottom boundary.

Press the buttons. Every face has the same base \(w\), but its shape decides what \(w\) is: a diameter, a side, a hypotenuse or a leg. Every other length follows from \(w\).

Notation

  1. The base width \(w(x)\) is the length of the face on the floor, the top boundary \(f(x)\) minus the bottom boundary \(g(x)\).

    \[\cy{w(x)} = f(x) - g(x)\]

  2. For a semicircle, \(w\) is the diameter, and the radius is half of it.

    \[\cy{r} = \frac{w}{2}\]

  3. For an equilateral triangle, \(w\) is a side. The height, half of the base and one slanted side form a right triangle, so \(h^2 + \left(\tfrac{w}{2}\right)^2 = w^2\) by the Pythagorean theorem.

    \[\cy{h} = \sqrt{w^2 - \tfrac{w^2}{4}} = \frac{\sqrt{3}}{2}\,w\]

  4. If a leg of a right triangle with equal legs lies on the floor, \(w\) is that leg. The other leg stands upright with the same length.

    \[\cy{h} = w\]

  5. If the hypotenuse lies on the floor, the two angles at the floor are 45°, because the legs are equal. The height then splits the face into two right triangles with equal legs, the height and half of \(w\).

    \[\cy{h} = \frac{w}{2}\]

When it fails

Two right triangles with equal legs can share the same base and still differ. Press the button to put a leg on the floor instead of the hypotenuse.

The base keeps its length, but the height doubles, because the upright leg is as long as the leg on the floor. Only the words of the problem say which side lies on the floor.

Example 1: a right triangle with its hypotenuse on the floor

The base is the region between \(y = x^2\) and \(y = 6 - x\). The cross sections are isosceles right triangles perpendicular to the x-axis, with the hypotenuse on the base. At \(x = 1\), find the hypotenuse, the height and the legs.

  1. At \(x = 1\), the line is at height 5 and the parabola at height 1. The hypotenuse is the orange segment on the floor, the difference of these heights.

    \[\cy{w(1)} = 5 - 1 = 4\]

  2. The height splits the face into two triangles with equal legs, so it is half of the hypotenuse.

    \[\cy{h} = \frac{4}{2} = 2\]

  3. The two legs are equal, so \(\text{leg}^2 + \text{leg}^2 = 4^2\) by the Pythagorean theorem, which gives \(\text{leg}^2 = 8\).

    \[\cy{\text{leg}} = \sqrt{8} = 2\sqrt{2}\]

Example 2: a semicircle

The base is the region between the x-axis and \(y = 6x - x^2\). The cross sections are semicircles perpendicular to the x-axis, with the diameter on the base. Find the radius at \(x = 2\).

  1. The lower boundary is the x-axis, so the diameter at \(x = 2\) is the height of the curve there.

    \[\cy{w(2)} = 6\cdot2 - 2^2 = 8\]

  2. A tempting wrong move: The base is 8 long, so the radius is 8.

    \[\cr{r = 8}\]

    A semicircle with radius 8 is 16 wide, so the red semicircle reaches 4 units past each end of the orange segment, outside the solid. That segment is the diameter, not the radius.

  3. The radius is half of the diameter.

    \[\cy{r} = \frac{8}{2} = 4\]

Practice

The base is the region between \(x = y^2\) and \(x = 6 - y\). The cross sections are equilateral triangles perpendicular to the y-axis. At \(y = 1\), find the side and the height.

The slices stand perpendicular to the y-axis, so the orange segment is horizontal. Drag the slice to \(y = 1\).

Solution

  1. At \(y = 1\), the line is at \(x = 5\) and the parabola is at \(x = 1\), so the side is the right value minus the left value.

    \[\cy{w(1)} = 5 - 1 = 4\]

  2. The height of an equilateral triangle is \(\tfrac{\sqrt3}{2}\) times its side.

    \[\cy{h} = \frac{\sqrt{3}}{2}\cdot 4 = 2\sqrt{3}\]

Questions

The base lies between the parabola \(y = x^2\) and the line \(y = 8\), with cross sections perpendicular to the x-axis. The orange segment is the base of the slice at \(x = 2\).

  1. The face is a semicircle whose diameter is the orange segment. What is its radius?

    • 2
    • 4
    • 8
    Answer

    2

  2. The face is an isosceles right triangle with one leg on the orange segment. What is the length of the other leg?

    • 4
    • 2
    • \(4\sqrt{2}\)
    Answer

    4

  3. The base lies between \(y = x^2\) and \(y = 5\). The cross sections are equilateral triangles perpendicular to the x-axis. What is the height of the triangle at \(x = 1\)?

    • \(2\sqrt{3}\)
    • \(4\sqrt{3}\)
    • 2
    Answer

    \(2\sqrt{3}\)

Summary

You read the length of the face on the floor, and every other length follows from it.

The shape of the face fixes each length: the radius is half the diameter, and the height of a triangle follows from its side.

Use this before you write the area of a triangular or semicircular face.

This is one of four skills in topic 8.8 of Which & Why: AP Calculus.