Find the indicated measure. 4. The exact radius of a circle with circumference 36 meters 5. The exact diameter of a circle with circumference 29 feet 6. The exact circumference of a circle with diameter 26 inches 7. The exact circumference of a circle with radius 15 centimeters
The Correct Answer and Explanation is:
4. The exact radius of a circle with circumference 36 meters
The formula for the circumference of a circle is given by: C=2πrC = 2\pi rC=2πr
Where:
- CCC is the circumference,
- rrr is the radius,
- π\piπ is approximately 3.1416.
To find the radius (rrr), we rearrange the formula: r=C2πr = \frac{C}{2\pi}r=2πC
Substituting C=36C = 36C=36 meters: r=362π=366.2832≈5.73 metersr = \frac{36}{2\pi} = \frac{36}{6.2832} \approx 5.73 \, \text{meters}r=2π36=6.283236≈5.73meters
So, the exact radius is 362π\frac{36}{2\pi}2π36 meters, or approximately 5.73 meters.
5. The exact diameter of a circle with circumference 29 feet
The formula for the circumference of a circle can also be written in terms of the diameter: C=πdC = \pi dC=πd
Where:
- CCC is the circumference,
- ddd is the diameter.
Rearranging the formula to solve for the diameter: d=Cπd = \frac{C}{\pi}d=πC
Substituting C=29C = 29C=29 feet: d=29π=293.1416≈9.24 feetd = \frac{29}{\pi} = \frac{29}{3.1416} \approx 9.24 \, \text{feet}d=π29=3.141629≈9.24feet
So, the exact diameter is 29π\frac{29}{\pi}π29 feet, or approximately 9.24 feet.
6. The exact circumference of a circle with diameter 26 inches
We use the formula for the circumference in terms of diameter: C=πdC = \pi dC=πd
Substituting d=26d = 26d=26 inches: C=π×26=3.1416×26≈81.68 inchesC = \pi \times 26 = 3.1416 \times 26 \approx 81.68 \, \text{inches}C=π×26=3.1416×26≈81.68inches
So, the exact circumference is π×26\pi \times 26π×26 inches, or approximately 81.68 inches.
7. The exact circumference of a circle with radius 15 centimeters
Again, using the formula for the circumference: C=2πrC = 2\pi rC=2πr
Substituting r=15r = 15r=15 centimeters: C=2π×15=30π centimetersC = 2\pi \times 15 = 30\pi \, \text{centimeters}C=2π×15=30πcentimeters
So, the exact circumference is 30π30\pi30π centimeters, or approximately 94.25 centimeters.
Summary:
- The exact radius of a circle with circumference 36 meters is 362π\frac{36}{2\pi}2π36 meters, or approximately 5.73 meters.
- The exact diameter of a circle with circumference 29 feet is 29π\frac{29}{\pi}π29 feet, or approximately 9.24 feet.
- The exact circumference of a circle with diameter 26 inches is π×26\pi \times 26π×26 inches, or approximately 81.68 inches.
- The exact circumference of a circle with radius 15 centimeters is 30π30\pi30π centimeters, or approximately 94.25 centimeters.
