Geometry and Trigonometry - Metric conversions

Convert $4.25 \text{ km}$ into centimeters ($cm$).

$4.25 \times 1,000 \times 100 = 425,000 \text{ cm}$

First, convert kilometers to meters by multiplying by $1,000$ ($4,250 \text{ m}$). Then, convert meters to centimeters by multiplying by $100$.

Calculate the area of a field in hectares ($ha$) if its area is $25,000 \text{ m}^2$.

$\frac{25,000}{10,000} = 2.5 \text{ ha}$

Since $1 \text{ ha} = 10,000 \text{ m}^2$, we divide the total square meters by $10,000$ to find the value in hectares.

A water tank has a volume of $0.8 \text{ m}^3$. Find its capacity in liters ($L$).

$0.8 \times 1,000 = 800 \text{ L}$

There are $1,000 \text{ liters}$ in $1 \text{ cubic meter}$. Therefore, multiply the volume in $m^3$ by $1,000$.

Convert $750 \text{ mm}^2$ to $\text{ cm}^2$.

$\frac{750}{10^2} = \frac{750}{100} = 7.5 \text{ cm}^2$

To convert from $mm^2$ to $cm^2$, divide by the square of the linear conversion factor ($10^2 = 100$).

1. Calculate volume in $\text{ m}^3$: $V = \text{Area} \times \text{depth}$ $V = 45 \times 2 = 90 \text{ m}^3$
2. Convert $\text{m}^3$ to liters using the factor $1 \text{ m}^3 = 1,000 \text{ L}$: $90 \times 1,000 = 90,000 \text{ L}$

First, the volume is found in cubic meters by multiplying the base area by height. Since $1 \text{ m}^3$ holds $1,000$ liters, we multiply the result by $1,000$.

1. Calculate area in $\text{cm}^2$: $A = 30 \times 30 = 900 \text{ cm}^2$
2. Convert to $\text{m}^2$ by dividing by $10,000$ (since $1 \text{ m}^2 = 100^2 \text{ cm}^2$): $A = \frac{900}{10,000} = 0.09 \text{ m}^2$

To convert square units, you must divide by the square of the linear conversion factor. Since there are $100 \text{ cm}$ in $1 \text{ m}$, there are $10,000 \text{ cm}^2$ in $1 \text{ m}^2$.