The problem describes a rectangle ABCD with a length of $30_8$ and a width of $100_6$. (a) The first part asks to convert the length and width to base 10. (b) The second part asks to calculate the perimeter of the rectangle in base 2.

1. Problem Description

The problem describes a rectangle ABCD with a length of $30_8$ and a width of $100_6$.

(a) The first part asks to convert the length and width to base

1

0. (b) The second part asks to calculate the perimeter of the rectangle in base

2.

2. Solution Steps

(a) Convert the length from base 8 to base 10:

$30_8 = 3 * 8^1 + 0 * 8^0 = 3 * 8 + 0 * 1 = 24 + 0 = 24_{10}$

Convert the width from base 6 to base 10:

$100_6 = 1 * 6^2 + 0 * 6^1 + 0 * 6^0 = 1 * 36 + 0 * 6 + 0 * 1 = 36 + 0 + 0 = 36_{10}$

(b) Calculate the perimeter in base 10:

Perimeter = 2 * (length + width)

Perimeter = 2 * (24 + 36)

Perimeter = 2 * 60

Perimeter = 120_{10}

Convert the perimeter from base 10 to base 2:

To convert 120 to base 2, we repeatedly divide by 2 and record the remainders:

120 / 2 = 60 remainder 0

60 / 2 = 30 remainder 0

30 / 2 = 15 remainder 0

15 / 2 = 7 remainder 1

7 / 2 = 3 remainder 1

3 / 2 = 1 remainder 1

1 / 2 = 0 remainder 1

Reading the remainders from bottom to top, we get $1111000_2$.

3. Final Answer

(a) Length = $24_{10}$, Width = $36_{10}$

(b) Perimeter = $1111000_2$