To find the surface area of a cone, we can use the formula:
\[ \text{Surface Area} = \pi r^2 + \pi r l \]
where:
- \(r\) is the radius of the base of the cone
- \(l\) is the slant height of the cone
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Calculate the radius (r): Given the diameter \(d = 25\) cm, the radius \(r\) is: \[ r = \frac{d}{2} = \frac{25}{2} = 12.5 \text{ cm} \]
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Calculate the slant height (l): The slant height can be calculated using the Pythagorean theorem. Given the height \(h = 22\) cm: \[ l = \sqrt{r^2 + h^2} \] Plugging in the values: \[ l = \sqrt{(12.5)^2 + (22)^2} = \sqrt{156.25 + 484} = \sqrt{640.25} \approx 25.25 \text{ cm} \]
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Calculate the surface area: Now we can substitute \(r\) and \(l\) into the surface area formula: \[ \text{Surface Area} = \pi r^2 + \pi r l \] \[ \text{Surface Area} = 3.14 \times (12.5)^2 + 3.14 \times 12.5 \times 25.25 \]
Calculating \( \pi r^2 \): \[ 3.14 \times (12.5)^2 = 3.14 \times 156.25 = 490.625 \]
Calculating \( \pi r l \): \[ 3.14 \times 12.5 \times 25.25 \approx 3.14 \times 316.25 \approx 995.325 \]
Now, adding these two areas together: \[ \text{Surface Area} = 490.625 + 995.325 = 1485.95 \text{ square centimeters} \]
Thus, the approximate surface area of the cone is 1485.95 square centimeters.
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