b) Find |AB|, 1) Convert 11010, to a number in base five, b) Arrange the following in ascending order 11010, 512 and 13,0, c) Evaluate 3125-143, give the answer in base 5

Here are the solutions to your questions:

: Vectors b) Find $|\vec{AB}|$. From the previous problem, $\vec{AB} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}$.

Step 1: Use the formula for the magnitude of a vector. The magnitude of a vector $\begin{pmatrix} x \\ y \end{pmatrix}$ is given by $\sqrt{x^2 + y^2}$. $|\vec{AB}| = \sqrt{3^2 + (-4)^2}$ Step 2: Calculate the squares and sum them. $|\vec{AB}| = \sqrt{9 + 16}$ $|\vec{AB}| = \sqrt{25}$ Step 3: Find the square root. $|\vec{AB}| = 5$ The magnitude of $\vec{AB}$ is $\boxed{5}$.

: Number Bases a) Convert $110101_2$ to a number in base five.

Step 1: Convert $110101_2$ to base 10. $110101_2 = 1 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0$ $= 1 \times 32 + 1 \times 16 + 0 \times 8 + 1 \times 4 + 0 \times 2 + 1 \times 1$ $= 32 + 16 + 0 + 4 + 0 + 1$ $= 53_{10}$ Step 2: Convert $53_{10}$ to base 5. Divide 53 by 5 repeatedly and record the remainders: $53 \div 5 = 10$ remainder 3 $10 \div 5 = 2$ remainder 0 $2 \div 5 = 0$ remainder 2 Reading the remainders from bottom to top, we get $203_5$. So, $110101_2$ in base five is $\boxed{203_5}$.

b) Arrange the following in ascending order: $110101_2$, $51_{10}$, and $13_{10}$.

Step 1: Convert all numbers to base 10 for comparison. From part a), $110101_2 = 53_{10}$. The other numbers are already in base 10: $51_{10}$ and $13_{10}$. Step 2: Arrange the base 10 numbers in ascending order. The numbers are $53_{10}$, $51_{10}$, $13_{10}$. In ascending order: $13_{10}$, $51_{10}$, $53_{10}$. Step 3: Write the numbers in their original bases. The ascending order is $\boxed{13_{10}, 51_{10}, 110101_2}$.

c) Evaluate $312_5 - 143_5$ and give the answer in base 5.

Step 1: Perform subtraction in base 5. $312_5$

• $143_5$

Starting from the rightmost digit: $2 - 3$: We need to borrow. Borrow 1 from the '1' in the middle column (which becomes '0'). The borrowed '1' is worth 5 in base 5. So, $2+5 = 7$. $7 - 3 = 4$.

Next column (middle): $0 - 4$: We need to borrow. Borrow 1 from the '3' in the leftmost column (which becomes '2'). The borrowed '1' is worth 5 in base 5. So, $0+5 = 5$. $5 - 4 = 1$.

Leftmost column: $2 - 1 = 1$.

Combining the results, we get $114_5$. $312_5 - 143_5 = \boxed{114_5}$

: Map Scale The scale on a map is given as $1:150\,000$.

a) Find the distance on the ground represented by 1 cm on the map. Give the answer in km.

Step 1: Interpret the scale. The scale $1:150\,000$ means that 1 cm on the map represents $150\,000$ cm on the ground. Step 2: Convert the ground distance from cm to km. We know that $1 \text{ km} = 1000 \text{ m}$ and $1 \text{ m} = 100 \text{ cm}$. So, $1 \text{ km} = 1000 \times 100 \text{ cm} = 100\,000 \text{ cm}$. Distance in km $= \frac{150\,000 \text{ cm}}{100\,000 \text{ cm/km}} = 1.5 \text{ km}$. The distance on the ground is $\boxed{1.5 \text{ km}}$.

b) Find the distance on the map representing a distance on the ground of 7.5 km. Give the answer in cm.

Step 1: Convert the ground distance from km to cm. Ground distance $= 7.5 \text{ km} = 7.5 \times 100\,000 \text{ cm} = 750\,000 \text{ cm}$. Step 2: Use the map scale to find the map distance. The scale is $1 \text{ cm (map)} : 150\,000 \text{ cm (ground)}$. Let $x$ be the distance on the map. $\frac{x \text{ cm (map)}}{750\,000 \text{ cm (ground)}} = \frac{1}{150\,000}$ $x = \frac{750\,000}{150\,000} \text{ cm}$ $x = 5 \text{ cm}$ The distance on the map is $\boxed{5 \text{ cm}}$.

c) Find the actual area represented by an area of 4 cm$^2$ on the map. Give the answer in km$^2$.

Step 1: Determine the area scale factor. The linear scale is $1:150\,000$. The area scale is $(1)^2 : (150\,000)^2 = 1 : 22\,500\,000\,000$. This means 1 cm$^2$ on the map represents $22\,500\,000\,000$ cm$^2$ on the ground. Step 2: Convert the ground area from cm$^2$ to km$^2$. We know that $1 \text{ km}^2 = (100\,000 \text{ cm})^2 = 10^{10} \text{ cm}^2$. So, 1 cm$^2$ (map) represents $\frac{22\,500\,000\,000 \text{ cm}^2}{10^{10} \text{ cm}^2/\text{km}^2} = 2.25 \text{ km}^2$ on the ground. Step 3: Calculate the actual area for 4 cm$^2$ on the map. Actual area $= 4 \times 2.25 \text{ km}^2 = 9 \text{ km}^2$. The actual area is $\boxed{9 \text{ km}^2}$.