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Find the volume of a right circular cone that has a height of 9.5cm and a base with a circumference of 9cm. Round your answer to the nearest tenth of a cubic centimeter. Answer: cm^(3)

Find the volume of a right circular cone that has a height of 9.5 cm 9.5 \mathrm{~cm} and a base with a circumference of 9 cm 9 \mathrm{~cm} . Round your answer to the nearest tenth of a cubic centimeter.\newlineAnswer: cm3 \mathrm{cm}^{3}

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Full solution

Q. Find the volume of a right circular cone that has a height of 9.5 cm 9.5 \mathrm{~cm} and a base with a circumference of 9 cm 9 \mathrm{~cm} . Round your answer to the nearest tenth of a cubic centimeter.\newlineAnswer: cm3 \mathrm{cm}^{3}
  1. Volume Formula for Cone: To find the volume of a cone, we need to know the radius of the base and the height. The formula for the volume of a cone is V=(13)πr2hV = (\frac{1}{3})\pi r^2 h, where rr is the radius and hh is the height. We are given the height as 9.59.5 cm, but we need to find the radius from the circumference of the base.
  2. Calculate Radius from Circumference: The circumference of a circle is given by the formula C=2πrC = 2\pi r, where CC is the circumference and rr is the radius. We can rearrange this formula to solve for the radius: r=C2πr = \frac{C}{2\pi}. The circumference is given as 99 cm.
  3. Calculate Volume with Radius and Height: Now we calculate the radius: r=9 cm(2π)r = \frac{9 \, \text{cm}}{(2\pi)}. Using π≈3.14159\pi \approx 3.14159, we get r≈9 cm(2×3.14159)≈9 cm6.28318≈1.432 cmr \approx \frac{9 \, \text{cm}}{(2 \times 3.14159)} \approx \frac{9 \, \text{cm}}{6.28318} \approx 1.432 \, \text{cm}.
  4. Perform Calculation: With the radius found, we can now calculate the volume of the cone using the volume formula V=13πr2hV = \frac{1}{3}\pi r^2h. Substituting the values we have r≈1.432r \approx 1.432 cm and h=9.5h = 9.5 cm, we get V≈13×3.14159×(1.432 cm)2×9.5 cmV \approx \frac{1}{3} \times 3.14159 \times (1.432\,\text{cm})^2 \times 9.5\,\text{cm}.
  5. Round to Nearest Tenth: Performing the calculation: V≈(1/3)×3.14159×(1.432 cm)2×9.5 cm≈(1/3)×3.14159×2.049824×9.5 cm≈20.431 cm3V \approx (1/3) \times 3.14159 \times (1.432 \, \text{cm})^2 \times 9.5 \, \text{cm} \approx (1/3) \times 3.14159 \times 2.049824 \times 9.5 \, \text{cm} \approx 20.431 \, \text{cm}^3.
  6. Final Volume: Finally, we round the answer to the nearest tenth of a cubic centimeter. The volume of the cone is approximately 20.4 cm320.4\,\text{cm}^3.

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