Question

order the set of numbers from least to greatest. then graph the set on a number line. 3 1/2, 10/3, π, √13. the radius of a circle can be approximated using the expression √(a/π) where a represents the area. a circular kiddie swimming - pool has an area of about 38 square feet. an inflatable full - size circular pool has an area of about 113 square feet. how much greater is the radius of the full - size pool than the radius of the kiddie pool? round to the nearest whole number. the time in seconds that it takes an object to fall d feet can be found using the expression √(d/16). in an egg drop contest, clara successfully dropped her egg container from a height of 35 feet, while vladimir successfully dropped his egg container from a height of 23 feet. how much longer did it take claras egg to reach the floor than vladimirs egg? round to the nearest tenth. which number is greater, 3.14 or π? justify your answer.

Explanation:

Step1: Convert mixed - number to improper fraction

$3\frac{1}{2}=\frac{3\times2 + 1}{2}=\frac{7}{2}=3.5$

Step2: Calculate the value of $\frac{10}{3}$

$\frac{10}{3}=10\div3\approx3.33$

Step3: Recall the value of $\pi$

$\pi\approx3.14$

Step4: Calculate the value of $\sqrt{13}$

Since $3^2 = 9$ and $4^2=16$, and $13$ is between $9$ and $16$, $\sqrt{13}\approx3.61$

Step5: Order the numbers

Comparing the values: $3.14\lt3.33\lt3.5\lt3.61$, so $\pi\lt\frac{10}{3}\lt3\frac{1}{2}\lt\sqrt{13}$

for circle - radius problem:

Step1: Recall the formula for radius in terms of area

The formula for the radius of a circle is $r = \sqrt{\frac{A}{\pi}}$.

Step2: Calculate the radius of the kiddie pool

For the kiddie pool with $A_1 = 38$ square feet, $r_1=\sqrt{\frac{38}{\pi}}\approx\sqrt{\frac{38}{3.14}}\approx\sqrt{12.09}\approx3.5$ feet.

Step3: Calculate the radius of the full - size pool

For the full - size pool with $A_2 = 113$ square feet, $r_2=\sqrt{\frac{113}{\pi}}\approx\sqrt{\frac{113}{3.14}}\approx\sqrt{36}\approx6$ feet.

Step4: Find the difference in radii

$r_2 - r_1=6 - 3.5 = 2.5\approx3$ feet

for egg - drop problem:

Step1: Recall the formula for time of free - fall

The time $t$ it takes for an object to fall $h$ feet is given by $t=\sqrt{\frac{h}{16}}$ (derived from $h = \frac{1}{2}gt^2$ with $g = 32$ ft/s²).

Step2: Calculate Clara's time

For Clara with $h_1 = 35$ feet, $t_1=\sqrt{\frac{35}{16}}\approx\sqrt{2.1875}\approx1.5$ seconds.

Step3: Calculate Vladimir's time

For Vladimir with $h_2 = 23$ feet, $t_2=\sqrt{\frac{23}{16}}\approx\sqrt{1.4375}\approx1.2$ seconds.

Step4: Find the difference in times

$t_1 - t_2=1.5 - 1.2 = 0.3$ seconds

for comparing $3.14$ and $\pi$:
$\pi$ is an irrational number with an approximate value of $3.14159\cdots$. The number $3.14$ is a rational approximation of $\pi$. Since $3.14159\cdots>3.14$, $\pi>3.14$.

Answer:

The numbers in order from least to greatest are $\pi,\frac{10}{3},3\frac{1}{2},\sqrt{13}$