Exact answers versus approximations

1. A square has area $23{\text{m}}^2$. Write its exact positive side.

   2 marks
2. Round $\sqrt{23}$ to two decimal places.

   2 marks
3. A circle has diameter $7\text{m}$. Give its exact circumference.

   2 marks
4. Round the circumference for diameter $7\text{m}$ to one decimal place.

   2 marks
5. A student uses $1.4$ for $\sqrt{2}$ and obtains $10\sqrt{2}\approx 14$. Find the result to two decimal places without rounding early.

   3 marks
6. For radius $50\text{cm}$ compare $C=100\pi$ with $100\left(3.14\right)$. How much smaller is the second value, to two decimal places?

   3 marks

Equivalent correct methods earn the corresponding marks. Include units where asked.

1. A square has area $23{\text{m}}^2$. Write its exact positive side.

   2 marks

   The positive number whose square is $23$ is $\sqrt{23}$.

   $\sqrt{23}\text{m}$.
2. Round $\sqrt{23}$ to two decimal places.

   2 marks

   $\sqrt{23}=4.795831\dots$; the third decimal rounds $4.79$ to $4.80$.

   $4.80$.
3. A circle has diameter $7\text{m}$. Give its exact circumference.

   2 marks

   $C=\pi d=7\pi \text{m}$.

   $7\pi \text{m}$.
4. Round the circumference for diameter $7\text{m}$ to one decimal place.

   2 marks

   $7\pi =21.991148\dots$, giving $22.0\text{m}$.

   $22.0\text{m}$.
5. A student uses $1.4$ for $\sqrt{2}$ and obtains $10\sqrt{2}\approx 14$. Find the result to two decimal places without rounding early.

   3 marks

   $10\sqrt{2}=14.142135\dots$.

   Round the complete calculation, not the root alone.

   $14.14$.
6. For radius $50\text{cm}$ compare $C=100\pi$ with $100\left(3.14\right)$. How much smaller is the second value, to two decimal places?

   3 marks

   $100\pi -314=0.159265\dots \text{cm}$.

   $0.16\text{cm}$ smaller.