Chapter 8.1, Problem 58E

(a)

To determine

The first five terms of the sequence.

Answer to Problem 58E

  $60,90,108,120,128.6$

Explanation of Solution

Given:A regular polygon has equal measures of side and length. For a regular $n$ sided polygon $(n\ge 3)$ the measure $a_n$ of an interior angle is given by

  $a_n=\frac{180(n-2)}{n}$

Given: $a_n=\frac{180(n-2)}{n}$

Since $(n\ge 3)$

The first five terms are

  $\begin{array}{l}{a}_3=\frac{180(3-2)}{3}=60\\ a_4=\frac{180(4-2)}{4}=90\\ a_5=\frac{180(5-2)}{5}=108\\ a_6=\frac{180(6-2)}{6}=120\\ a_7=\frac{180(7-2)}{7}=128.6\end{array}$

Hence, $60,90,108,120,128.6$

(b)

To determine

To Find:The rule for sequence giving the sum $T_n$ of the measures of the interior angles of the polygon.

Answer to Problem 58E

  $T_n=180(n-2)$

Explanation of Solution

Given:A regular polygon has equal measures of side and length. For a regular $n$ sided polygon $(n\ge 3)$ the measure $a_n$ of an interior angle is given by

  $a_n=\frac{180(n-2)}{n}$

The rule giving the sum is

  $\begin{array}{l}{T}_n=\frac{180(n-2)}{n}\times n\\ T_n=180(n-2)\end{array}$

Hence, $T_n=180(n-2)$

(c)

To determine

To Find:The sum $T_n$ of the measures of the interior angles of Guggenheim Museum.

Answer to Problem 58E

  $T_{12}=1800$

Explanation of Solution

Given: A regular polygon has equal measures of side and length. For a regular $n$ sided polygon $(n\ge 3)$ the measure $a_n$ of an interior angle is given by

  $a_n=\frac{180(n-2)}{n}$

  

From the figure it is seen that $n=12$

Substitute

  $n=12$ in $T_n=180(n-2)$

We get

  $\begin{array}{l}{T}_{12}=180(12-2)\\ T_{12}=1800\end{array}$