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A hemispherical bowl is made of steel, 0...

A hemispherical bowl is made of steel, 0.5 cm thick. The inside radius of the bowl is 4 cm. Find the volume of steel used in making the bowl.

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To find the volume of steel used in making the hemispherical bowl, we will follow these steps: ### Step 1: Identify the given values - Inner radius of the bowl (r) = 4 cm - Thickness of the bowl = 0.5 cm ### Step 2: Calculate the outer radius The outer radius (R) can be calculated by adding the thickness to the inner radius: \[ R = r + \text{thickness} = 4 \, \text{cm} + 0.5 \, \text{cm} = 4.5 \, \text{cm} \] ### Step 3: Calculate the volume of the outer hemisphere The formula for the volume of a hemisphere is given by: \[ V = \frac{2}{3} \pi R^3 \] Substituting the value of the outer radius: \[ V_{\text{outer}} = \frac{2}{3} \pi (4.5)^3 \] Calculating \( (4.5)^3 \): \[ (4.5)^3 = 91.125 \] Now substituting this back into the volume formula: \[ V_{\text{outer}} = \frac{2}{3} \pi (91.125) \] ### Step 4: Calculate the volume of the inner hemisphere Using the same formula for the inner hemisphere: \[ V_{\text{inner}} = \frac{2}{3} \pi r^3 \] Substituting the value of the inner radius: \[ V_{\text{inner}} = \frac{2}{3} \pi (4)^3 \] Calculating \( (4)^3 \): \[ (4)^3 = 64 \] Now substituting this back into the volume formula: \[ V_{\text{inner}} = \frac{2}{3} \pi (64) \] ### Step 5: Calculate the volume of steel used The volume of steel used in making the bowl is the difference between the outer and inner volumes: \[ V_{\text{steel}} = V_{\text{outer}} - V_{\text{inner}} \] Substituting the values we found: \[ V_{\text{steel}} = \frac{2}{3} \pi (91.125) - \frac{2}{3} \pi (64) \] Factoring out \( \frac{2}{3} \pi \): \[ V_{\text{steel}} = \frac{2}{3} \pi (91.125 - 64) \] Calculating \( 91.125 - 64 \): \[ 91.125 - 64 = 27.125 \] So, \[ V_{\text{steel}} = \frac{2}{3} \pi (27.125) \] ### Step 6: Final calculation Using \( \pi \approx \frac{22}{7} \): \[ V_{\text{steel}} = \frac{2}{3} \times \frac{22}{7} \times 27.125 \] Calculating: \[ V_{\text{steel}} \approx \frac{44}{21} \times 27.125 \] Calculating \( \frac{44 \times 27.125}{21} \): \[ V_{\text{steel}} \approx 56.213 \, \text{cm}^3 \] ### Final Answer The volume of steel used in making the bowl is approximately \( 56.213 \, \text{cm}^3 \). ---
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