Bot GPT 4-o mini To find the surface area of the entire cone, we need to calculate both the lateral surface area and the base area of the cone.
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Calculate the radius of the cone: The diameter of the cone is 16 inches, so the radius \( r \) is: \[ r = \frac{\text{diameter}}{2} = \frac{16}{2} = 8 \text{ inches} \]
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Calculate the lateral surface area of the cone: The formula for the lateral surface area \( A_L \) of a cone is: \[ A_L = \pi r l \] where \( l \) is the slant height. Given \( l = 22 \) inches: \[ A_L = 3.14 \times 8 \times 22 \] \[ A_L = 3.14 \times 176 = 552.64 \text{ square inches} \]
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Calculate the base area of the cone: The formula for the area of the base \( A_B \) (which is a circle) is: \[ A_B = \pi r^2 \] \[ A_B = 3.14 \times 8^2 = 3.14 \times 64 = 200.96 \text{ square inches} \]
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Calculate the total surface area of the cone: The total surface area \( A_{total} \) is the sum of the lateral surface area and the base area: \[ A_{total} = A_L + A_B \] \[ A_{total} = 552.64 + 200.96 = 753.60 \text{ square inches} \]
Thus, the surface area of the entire cone peak, rounded to the nearest hundredth, is: \[ \boxed{753.60} \text{ square inches} \]
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