All About Pi

All About Pi exercise

• Identify the different parts of a circle.

  Hints

  The diameter and radius go through the center of the circle.

  The area and circumference are calculations of the space inside the circle and the distance around the circle, respectively.

  The diameter is $8~\text{cm}$ and the radius is $4~\text{cm}$.

  Solution

  Center

  • The center is the point in the middle of the circle.

  Radius

  • The radius is the line that extends from the center to the edge.

  Diameter

  • The diameter is the line that extends from one side of the circle to the other, passing through the center.

  Circumference

  • The circumference is a measure of the distance around the circle.

  Area

  • The area is the total amount of space inside of the circle.

• Calculate area and circumference of the given circles.

  Hints

  The formula for the area of a circle is, $A=\pi r^2$.

  The formula for the circumference of a circle is, $c=2\pi r$.

  If the radius of the circle is 10cm,

  $\text{Area}=\pi r^2=\pi (10)^2=100 (3.14)= 314$

  $\text{Circumference}=2\pi r=2(3.14)(10)=62.8$

  If the diameter of the circle is 10cm,

  $\mathrm{Area}=\pi r^2=\pi (5)^2=25 (3.14)= 78.5$

  $\mathrm{Circumference}=d\pi =10\pi=31.4$

  Solution

  1. Area

  The area of a pie can be calculated by using the formula, $A=\pi r^2$, where $r$ represents the radius of the circle.

  To solve for the area of the pie, substitute 5 into the equation for $r$, and simplify.

  • $A=\pi(5^2)=25\pi$

  $\pi $ or pi, is an irrational number. This means that $\pi$ has an infinite decimal expansion. Instead of writing all of the digits for pi, the common approximation used for the irrational number is $3.14$.

  This means that the answer can be written as,

  • $A=25(3.14)=78.5$

  2. Circumference

  The circumference of a pie can be calculated by using the formula, $c=2\pi r$ where $r$ represents the radius of the circle.

  Since the radius is twice the diameter or $2r$, another way to write the circumference formula is, $c=d\pi$.

  We are given that the radius is $6$cm. This means that the diameter of the circle is $12$cm.

  Therefore, we can represent the circumference with the radius formula, $c=2\pi (6)$ which is equivalent to the the diameter formula, $C=12\pi$.

  We can substitute $\pi$ and find the circumference of the circle to be, $c=12(3.14)=37.68$.

  3. The area of the cherry pie with radius 7cm is,

  • $\mathrm{Area}=\pi r^2=\pi (7^2)=49\pi$ or $153.86$

  4. The circumference of the apple pie with diameter 6cm is,

  • Using the diameter formula, $\mathrm{circumference}=d\pi =6\pi$ or $18.84$
  • Using the radius formula, $\mathrm{circumference}=2\pi r=2\pi (3)=6\pi$ or $18.84$

• Identify true statements about circles and $\pi$.

  Hints

  A circle with a radius of $4$ has an area of $16\pi$.

  A circle with a radius of $4$ has a circumference of $8\pi $.

  A circle with a diameter of $6$ has an area of $9\pi$ and a circumference of $6\pi$.

  Solution

  True Statements

  • The area of a circle is $A=\pi r^2$.
  • The radius extends from the center to the edge.
  • Circumference is the distance around the edge of a circle.
  • $\pi =3.14$
  • $2$ times the radius is the diameter.

  False Statements

  • $\pi$ is a rational number is false because $\pi$ is an irrational number. An irrational number cannot be expressed as a ratio between two numbers or as a simple fraction because there is not a finite number of numbers when written as a decimal. Instead, the numbers in the decimal expansion go on forever, without repeating. This is why $\pi$, which equals 3.14159265..., is an irrational number.
  • The diameter of a circle is half of the radius is false because the radius of a circle is half of the diameter.
  • The circumference of a circle is, $2\pi d$ is false because the circumference formula is $c=2\pi r$. Since the radius of a circle is half the diameter, the circumference formula can also be written as $d\pi$ but not $2\pi d$.

• Calculate the distance traveled by the delivery truck.

  Hints

  Let $r$ represent the radius of a circle.

  • $\text{revolutions}=\frac{\text{distance}}{2\pi r}$

  A delivery truck has wheels with a radius 13.125 inches. The truck drives 20 feet to the first house and 42 feet to the second house. Find the number of rotations:

  • Add the two distances together: $20+42=62$ feet
  • Since the radius is in inches, convert $62$ feet to inches: $62(12)=744$ inches
  • $\text{revolutions}=\frac{\text{distance}}{2\pi r}$
  • $\text{revolutions}=\frac{744}{2(3.14)(13.125)}$
  • $\text{revolutions}=\frac{744}{82.43}$
  • $\text{revolutions}=9$
  • Each wheel makes $9$ revolutions.

  A delivery truck has wheels with a radius $13.25$ inches. Each wheel makes $17$ rotations during the delivery. Find the distance of the trip:

  • $\mathrm{revolutions}=\frac{\mathrm{distance}}{2\pi r}$
  • $17=\frac{\mathrm{distance}}{2(3.14)(13.125)}$
  • $17=\frac{\mathrm{distance}}{82.43}$
  • $17(82.43)=\mathrm{distance}$
  • $\mathrm{distance}=1,401.31$ inches

  To convert the distance to feet, divide by $12$:

  • $\frac{1401.31}{12}=116.77$

  Therefore, the delivery truck traveled about $117$ feet.

  Solution

  For the following problems,

  • The radius of each wheel is $15.915$ inches.
  • Revolutions and rotations mean the same thing.
  • $1\text{ft}=12\text{in}$

  1. The owner is driving $84$ feet to deliver $5$ pies. How many revolutions does each wheel make?

  • $\text{revolutions}=\frac{\text{distance}}{2\pi r}$
  • Since the radius is in inches, convert $84$ feet to inches: $84(12)=1008$ inches
  • revolutions$=\frac{1008}{2(3.14)(15.915)}$
  • revolutions$=\frac{1008}{99.95}$
  • revolutions$=10.08$
  • Each wheel makes about $10$ rotations.

  2. The owner is making two trips. First, he drives $1250$ feet to the first house, then another $90$ feet to the next house. How many revolutions does each wheel make for the total trip?

  • Add the two distances together: $1250+90=1340$ft
  • $\text{revolutions}=\frac{\text{distance}}{2\pi r}$
  • Since the radius is in inches, convert $1340$ feet to inches: $1340(12)=16080$ inches
  • revolutions$=\frac{16080}{2(3.14)(15.915)}$
  • revolutions$=\frac{16080}{99.95}$
  • revolutions$=160.88$
  • Each wheel makes about $161$ rotations.

  3. If each wheel makes $125$ rotations during one trip, what is the total distance for the delivery?

  • $\text{revolutions}=\frac{\text{distance}}{2\pi r}$
  • $125=\frac{\text{distance}}{2(3.14)(15.915)}$
  • $125=\frac{\text{distance}}{99.95}$
  • $125(99.95)=\text{distance}$
  • $\text{distance}=12493.75$ inches
  • To convert the distance to feet, divide by $12$: $\frac{12,493.75}{12}=1041.15$
  • Therefore, the delivery truck traveled about $1041$ feet.

  4. One Sunday, the owner made three house deliveries. The following wheel rotations were required for each trip:

  • $14$ rotations
  • $75$ rotations
  • $47$ rotations

  What is the total distance for all three deliveries?

  • Add the rotations together: $14+75+47=136$
  • $\mathrm{revolutions}=\frac{\mathrm{distance}}{2\pi r}$
  • $136=\frac{\mathrm{distance}}{2(3.14)(15.915)}$
  • $136=\frac{\mathrm{distance}}{99.95}$
  • $136(99.95)=\mathrm{distance}$
  • $\mathrm{distance}=13595.2$ inches
  • To convert the distance to feet, divide by $12$: $\frac{13593.2}{12}=1132.77$
  • Therefore, the delivery truck traveled about $1133$ feet.

• Determine which numbers are rational and which numbers are irrational.

  Hints

  A rational number can be written as a fraction where both the numerator and denominator are integers.

  A irrational number cannot be expressed as a fraction.

  Rational Numbers:

  • $5$, $\frac{1}{2}$, $-2$, $-\frac{7}{8}$

  Irrational Numbers:

  • $0.07007...$, $0.78788...$

  Solution

  A rational number is a number that can be written as a fraction where both the numerator and denominator are integers, and the denominator is not zero:

  • $\frac{1}{9}$, $-5$, $-\frac{2}{3}$, $-0.275$, $0.333333$

  An irrational number cannot be expressed as a fraction between two integers. Instead, the decimal expansion goes on forever:

  • $\pi$, $0.03003...$, $0.347892...$, $2.71828...$, $3.1415926...$

• Determine the different measurements of the Ferris wheel.

  Hints

  Let $r$ represent the radius of a circle.

  • $\text{Area}=\pi r^2$
  • $\text{Circumference}=2\pi r$
  • $ \text{Revolutions}=\frac{\text{Distance}}{2\pi r}$

  A circle with diameter $10$ has the following area and circumference:

  • $\text{Area}=3.14(5^2)=78.5$.
  • $\text{Circumference}=2(3.14)(5)=314$.

  A wheel with radius $2~\text{ft}$ and that travels $100~\text{ft}$ has the following number of rotations:

  • $\text{revolutions}=\frac{100}{2(3.14)(2)}$
  • $\text{revolutions}=\frac{100}{12.56}$
  • $\text{revolutions}=7.96$
  • The wheel makes about $8$ rotations.

  Solution

  1. The radius of the Ferris wheel is $39$ feet.

  • The radius of a circle is half of the diameter.
  • Since the diameter is $78$ft, $\frac{1}{2}(78)=39$.

  2. The circumference of the Ferris wheel is $245$ feet.

  • The formula for circumference is, $C=2\pi r$.
  • $c=2(3.14)(39)$
  • $c=244.92$
  • Rounding $244.92$ to the nearest whole number yields $245$.

  3. The area of the Ferris wheel is $4776$ feet squared.

  • The formula for area is, $A=\pi r^2$.
  • $A=3.14(39^2)$
  • $A=4775.94$
  • Rounding $4775.94$ to the nearest whole number yields $4776$.

  4. If the Ferris wheel travels $745$ feet, how many rotations did it make?

  • $\text{revolutions}=\frac{\text{distance}}{2\pi r}$
  • Since both the distance and radius are measured in feet, we do not have to do any unit conversions.
  • revolutions$=\frac{745}{2(3.14)(39)}$
  • revolutions$=\frac{745}{244.92}$
  • revolutions$=3.04$
  • Each wheel makes about $3$ rotations.

  5. If the Ferris wheel makes $5$ rotations, how many feet did it travel?

  • $\text{revolutions}=\frac{\text{distance}}{2\pi r}$
  • $5=\frac{\text{distance}}{2(3.14)(39)}$
  • $5=\frac{\text{distance}}{244.92}$
  • $5(244.92)=\text{distance}$
  • $\text{distance}=1224.6$inches
  • Therefore, the delivery truck traveled about $1225$ feet.