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INVESTIGATION ESTIMATING π
This investigation is about using relative frequency to estimate the value of π.
Lee draws circles on rectangular pieces of paper. He drops grains of rice at random onto the pieces of paper. He counts the number of grains of rice inside each circle.
Lee draws a circle of radius 5 cm on a rectangular piece of paper measuring 40 cm by 20 cm.
(a) (i) Find the area of the rectangle.
(ii) The probability, $p$, that a grain of rice lands inside the circle is $$ p = \frac{\text{area of circle}}{\text{area of rectangle}} $$
The area of the circle is $25 \times \pi$.
Show that $p$ is approximately 0.098 for this piece of paper.
(b) Lee drops 10 grains of rice at random onto the piece of paper. The diagram shows the result.
(i) How many of the grains of rice are inside the circle?
(ii) The relative frequency that a grain of rice is inside the circle $$ = \frac{\text{number of grains of rice inside the circle}}{\text{total number of grains of rice dropped}} $$
Find the relative frequency that a grain of rice is inside the circle.
(c) Lee drops 10 more grains of rice at random onto the piece of paper.
Show that the relative frequency that a grain of rice is inside the circle is 0.15.
(d) The relative frequency that a grain of rice is inside the circle gives an estimate for the probability, $p$.
The area of the circle is $25 \times \pi$.
Use $$ \frac{\text{area of circle}}{\text{area of rectangle}} = 0.15 $$ to show that an estimate for $\pi$ is 4.8.
597
546
562
Lee draws a circle of radius 10 cm on a rectangular piece of paper measuring 30 cm by 20 cm.
(a) Complete this statement with a number.
Area of circle = .................... \times \pi
(b) Lee drops 10 grains of rice at random onto the piece of paper. Diagram A shows the result.
Lee removes the 10 grains of rice and drops another 10 grains of rice at random onto the piece of paper. Diagram B shows the result.
(i) Complete the table.
| | A | B | Combined results for all 20 grains of rice |
|------------|---|---|--------------------------------------------|
| Number of grains of rice inside circle | | | 10 |
| Relative frequency | | | \frac{10}{20} |
(ii) Use the formula
$$\frac{\text{area of circle}}{\text{area of rectangle}} = \text{relative frequency}$$
to estimate \pi using the combined results for all 20 grains of rice.
\pi = ................................................
518
528
594
Lee draws two circles on a different rectangular piece of paper. The circles touch the edges of the piece of paper and touch each other.
(a) (i) Find the value of $x$.
.........................................................
(ii) Complete this statement with a number.
Total area of the two circles $= ................. \times \pi$
(b) Lee drops 50 grains of rice at random onto the piece of paper. He removes the 50 grains of rice and drops another 50 grains of rice at random onto the piece of paper. The table shows his results.
[Table_1]
Complete the table.
(c) Use the formula $ \frac{\text{total area of circles}}{\text{area of rectangle}} = \text{relative frequency}$ to estimate $\pi$.
$\pi = .......................................................$
(d) Give one reason why the estimate for $\pi$ in question 3(c) is more accurate than the estimate for $\pi$ in question 2(b)(ii).
..........................................................................................................................................................
573
549
521
Lee draws one circle of radius $r$ cm on a different piece of paper. The circle touches all four edges of the paper.
(a) Lee drops 500 grains of rice at random onto the piece of paper. He removes the 500 grains of rice and drops another 500 grains of rice at random onto the piece of paper. The combined number of grains of rice inside the circle is 785.
Use the formula $\frac{\text{area of circle}}{\text{area of rectangle}} = \text{relative frequency}$ to estimate $\pi$.
$\pi = \text{............................................}$
(b) Complete this sentence with a single number.
To estimate the value of $\pi$ when the circle touches all four edges of the paper, multiply the relative frequency by $\text{.......................}$
(c) Lee drops $n$ grains of rice at random onto the piece of paper. He removes the $n$ grains of rice and drops another $n$ grains of rice at random onto the piece of paper. The combined number of grains of rice inside the circle is $k$.
Show that $\frac{2k}{n}$ is an estimate for $\pi$.
582
517
593
