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What is the volume of a hemisphere with a diameter of 52.5ft, rounded to the nearest tenth of a cubic foot? Answer: ft^(3)

What is the volume of a hemisphere with a diameter of 52.5ft 52.5 \mathrm{ft} , rounded to the nearest tenth of a cubic foot?\newlineAnswer: ft3 \mathrm{ft}^{3}

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Full solution

Q. What is the volume of a hemisphere with a diameter of 52.5ft 52.5 \mathrm{ft} , rounded to the nearest tenth of a cubic foot?\newlineAnswer: ft3 \mathrm{ft}^{3}
  1. Calculate Radius: To find the volume of a hemisphere, we first need to find the volume of a full sphere and then divide it by 22. The formula for the volume of a sphere is V=43πr3V = \frac{4}{3}\pi r^3, where rr is the radius of the sphere. Given the diameter of the sphere is 52.552.5 feet, we can find the radius by dividing the diameter by 22.
  2. Calculate Volume of Full Sphere: Calculate the radius of the sphere. The radius rr is half of the diameter, so r=diameter2=52.5ft2=26.25ftr = \frac{\text{diameter}}{2} = \frac{52.5\text{ft}}{2} = 26.25\text{ft}.
  3. Calculate Numerical Value: Now, we can calculate the volume of the full sphere using the formula V=43πr3V = \frac{4}{3}\pi r^3. Substituting the value of r=26.25r = 26.25ft, we get V=43π(26.25ft)3V = \frac{4}{3}\pi (26.25\text{ft})^3.
  4. Divide Volume by 22: Perform the calculation for the volume of the full sphere. V=(43)π(26.25 ft)3=(43)π(26.25 ft×26.25 ft×26.25 ft)V = \left(\frac{4}{3}\right)\pi(26.25\,\text{ft})^3 = \left(\frac{4}{3}\right)\pi(26.25\,\text{ft} \times 26.25\,\text{ft} \times 26.25\,\text{ft}).
  5. Calculate Final Volume: Calculate the numerical value for the volume of the full sphere. V=43π(26.25 ft×26.25 ft×26.25 ft)≈43π(18150.625 ft3)≈43×π×18150.625 ft3≈24134.375π ft3V = \frac{4}{3}\pi(26.25\,\text{ft} \times 26.25\,\text{ft} \times 26.25\,\text{ft}) \approx \frac{4}{3}\pi(18150.625\,\text{ft}^3) \approx \frac{4}{3} \times \pi \times 18150.625\,\text{ft}^3 \approx 24134.375\pi\,\text{ft}^3.
  6. Calculate Numerical Value: To find the volume of the hemisphere, divide the volume of the full sphere by 22. Vhemisphere=Vsphere2≈(24134.375πft3)2≈12067.1875πft3V_{\text{hemisphere}} = \frac{V_{\text{sphere}}}{2} \approx \frac{(24134.375\pi \text{ft}^3)}{2} \approx 12067.1875\pi \text{ft}^3.
  7. Perform Final Calculation: Now, we need to calculate the numerical value for the volume of the hemisphere. Using the approximation π≈3.14159\pi \approx 3.14159, we get Vhemisphere≈12067.1875×3.14159 ft3V_{\text{hemisphere}} \approx 12067.1875 \times 3.14159\,\text{ft}^3.
  8. Round Volume: Perform the final calculation for the volume of the hemisphere. Vhemisphere≈12067.1875×3.14159 ft3≈37909.511 ft3V_{\text{hemisphere}} \approx 12067.1875 \times 3.14159\,\text{ft}^3 \approx 37909.511\,\text{ft}^3.
  9. Round Volume: Perform the final calculation for the volume of the hemisphere. Vhemisphere≈12067.1875×3.14159 ft3≈37909.511 ft3V_{\text{hemisphere}} \approx 12067.1875 \times 3.14159\,\text{ft}^3 \approx 37909.511\,\text{ft}^3.Round the volume of the hemisphere to the nearest tenth of a cubic foot. The rounded volume is approximately 37909.5 ft337909.5\,\text{ft}^3.

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