Chapter 9, Problem 68RE

$\text{(a)}$

To determine

To find: The infinite series.

Answer to Problem 68RE

The answer: ${\displaystyle \sum _{n=1}^{\infty }{\left(\frac{3}{4}\right)}^{n-1}}\frac{b^2\sqrt{3}}{4}$

Explanation of Solution

Given information:

The figure is:

  

Calculation:

The area of the triangle is:

  $\frac{{(2b)}^2\sqrt{3}}{4}=b^2\sqrt{3}$

At each step, take into account the deleted triangles' surface area. Due to the removed triangle's side being bb, the area of the single removed triangle in the first step is exactly a quarter of the area of the original triangle. To put it another way, this region.

  $\frac{b^2\sqrt{3}}{4}$

The size of the three new triangles that are removed in the second step is three times greater than the area of the triangle that was removed in the first step, i.e.

  $\frac{3b^2\sqrt{3}}{4^2}$

The area of the removed triangles in the subsequent stage is always three times the quarter of the area of the triangles that were removed in the preceding step.

  ${\displaystyle \sum _{n=1}^{\infty }{\left(\frac{3}{4}\right)}^{n-1}}\frac{b^2\sqrt{3}}{4}$

Therefore, the required infinite series is:

  ${\displaystyle \sum _{n=1}^{\infty }{\left(\frac{3}{4}\right)}^{n-1}}\frac{b^2\sqrt{3}}{4}$

$\text{(b)}$

To determine

To find: The sum of the infinite series and total area removed from the original triangle.

Answer to Problem 68RE

The answer:

The total area removed from the original triangle is $b^2\sqrt{3}$ .

Explanation of Solution

Given information:

The figure is:

  

Calculation:

From part $\text{(a)}$ a geometric series with

  $\begin{array}{c}a=b^2\sqrt{3}/4\\ r=3/4<1\end{array}$

Since,

  $\begin{array}{c}{\displaystyle \sum _{n=1}^{\infty }{\left(\frac{3}{4}\right)}^{n-1}}\frac{b^2\sqrt{3}}{4}=\frac{\frac{b^2\sqrt{3}}{4}}{1-\frac{3}{4}}\\ =b^2\sqrt{3}\end{array}$

Therefore, the required total area removed from the original triangle is $b^2\sqrt{3}$ .

$(c)$

To determine

To find: The every point on the original triangle removed or not.

Answer to Problem 68RE

The answer: No

Explanation of Solution

Given information:

The figure is:

  

Calculation:

For instance, consider the sides of the triangle, always remove points whose distance from the ends of the given side is a rational number. More precisely, $k\cdot b/2^n$ for appropriate integers $k$ and $n(k\le n)$ .

As a result, those points will never be eliminated if their distance from the side's end has an unreasonable value.

Therefore, the required no every point on the original triangle removed.