For a circle that has a diameter of 54 inches, what is the circumference in feet and inches? Give your answer to the nearest 1/8 inch: (12 inches = 1 foot) 102-2.6-8: Find the radius of a circle that has a circumference of 841 cm. 102-2.6-9: Find the area of a circle, in terms of π, that has a radius of 6 inches.
The Correct Answer and Explanation is:
102-2.6-7: Circumference of a Circle with Diameter 54 Inches
The formula for the circumference CCC of a circle is:C=π×dC = \pi \times dC=π×d
Given diameter d=54d = 54d=54 inches:C=π×54≈3.1416×54=169.6464 inchesC = \pi \times 54 \approx 3.1416 \times 54 = 169.6464 \text{ inches}C=π×54≈3.1416×54=169.6464 inches
Now convert this to feet and inches:
- There are 12 inches in a foot.
- Divide 169.6464169.6464169.6464 by 12:
169.6464÷12=14 feet and 1.6464 inches169.6464 \div 12 = 14 \text{ feet and } 1.6464 \text{ inches}169.6464÷12=14 feet and 1.6464 inches
To convert the decimal inches to the nearest 1/8 inch:
- Multiply 0.6464 by 8:
0.6464×8=5.17120.6464 \times 8 = 5.17120.6464×8=5.1712
Rounded to the nearest whole number: 5
So, the remaining fraction is 58\frac{5}{8}85 inch.
Answer: 14 feet 158 inches\boxed{14 \text{ feet } 1 \frac{5}{8} \text{ inches}}14 feet 185 inches
102-2.6-8: Radius from Circumference 841 cm
Use the circumference formula:C=2πrC = 2\pi rC=2πr
Given C=841C = 841C=841 cm:r=C2π=8412×3.1416≈8416.2832≈133.84 cmr = \frac{C}{2\pi} = \frac{841}{2 \times 3.1416} \approx \frac{841}{6.2832} \approx 133.84 \text{ cm}r=2πC=2×3.1416841≈6.2832841≈133.84 cm
Answer: 133.84 cm\boxed{133.84 \text{ cm}}133.84 cm
102-2.6-9: Area of Circle in Terms of π\piπ, Radius 6 Inches
The area AAA of a circle is:A=πr2A = \pi r^2A=πr2
Given radius r=6r = 6r=6 inches:A=π×62=π×36A = \pi \times 6^2 = \pi \times 36A=π×62=π×36
Answer: 36π in2\boxed{36\pi \text{ in}^2}36π in2
Explanation
The circumference of a circle depends directly on its diameter. The relationship is expressed through the formula C=πdC = \pi dC=πd, where π\piπ represents the mathematical constant approximately equal to 3.1416. When a diameter of 54 inches is used, multiplying by π\piπ gives an approximate circumference of 169.6464 inches. Converting to feet involves dividing the total inches by 12, yielding 14 full feet and a remainder of approximately 1.6464 inches. This remaining portion is converted to eighths by multiplying the decimal by 8 and rounding to the nearest whole number. This gives a final answer in mixed units—feet and inches—precise to the nearest eighth inch.
When reversing the process—finding the radius from a given circumference—the circumference formula is rearranged to isolate the radius: r=C2πr = \frac{C}{2\pi}r=2πC. Substituting 841 cm for the circumference and dividing by 2π2\pi2π gives a radius of approximately 133.84 cm. This shows how circumference and radius scale proportionally in circular geometry.
To find the area of a circle in terms of π\piπ, the expression A=πr2A = \pi r^2A=πr2 is used. For a radius of 6 inches, squaring the radius gives 36, which is then multiplied by π\piπ. The area remains in terms of π\piπ for exactness, avoiding any rounding errors associated with decimal approximations. This approach is particularly useful in algebraic or symbolic expressions.
