What is the surface area of the cone? Use 3.14 for pi(1 point)
the base is d=25cm and the height is 22cm
1. 2,383.26
2. 3,689.5 square centime
To find the surface area of a cone, you need to calculate both the base area and the lateral surface area (the area of the side of the cone).
The base of the cone is a circle, and its area (A_base) can be calculated using the formula:
\[ A_{\text{base}} = \pi \times r^2 \]
Given the diameter (d) is 25 cm, the radius (r) is half of that, so r = 25 cm / 2 = 12.5 cm.
Calculating the base area:
\[ A_{\text{base}} = \pi \times (12.5 \text{ cm})^2 = 3.14 \times 156.25 \text{ cm}^2 = 490.625 \text{ cm}^2 \]
Next, we need to calculate the lateral surface area (A_lateral). For a cone, the lateral surface area is the product of pi, the radius, and the slant height (l). The slant height can be found using the Pythagorean theorem, since it's the hypotenuse of the right-angled triangle formed by the radius of the base, the height of the cone, and the slant height itself:
\[ l = \sqrt{r^2 + h^2} \]
Here, h is the height of the cone, which is given as 22 cm. So let's calculate the slant height (l):
\[ l = \sqrt{(12.5 \text{ cm})^2 + (22 \text{ cm})^2} = \sqrt{156.25 \text{ cm}^2 + 484 \text{ cm}^2} = \sqrt{640.25 \text{ cm}^2} = 25.3 \text{ cm} \]
Now, we can calculate the lateral surface area:
\[ A_{\text{lateral}} = \pi \times r \times l = 3.14 \times 12.5 \text{ cm} \times 25.3 \text{ cm} = 3.14 \times 316.25 \text{ cm}^2 ≈ 993.375 \text{ cm}^2 \]
Finally, to find the total surface area (A_total) of the cone, we combine the base area and the lateral surface area:
\[ A_{\text{total}} = A_{\text{base}} + A_{\text{lateral}} = 490.625 \text{ cm}^2 + 993.375 \text{ cm}^2 = 1484 \text{ cm}^2 \]
None of the provided options match the calculated surface area:
1. 2,383.26 cm²
2. 3,689.5 cm²
It appears there might be a miscalculation or a confusion with the given options. The calculated surface area for the cone should be approximately 1484 cm². Please check the provided options or the calculations again.
What is the surface area of the cone? Use 3.14 for pi(1 point)
the base is d=25cm and the height is 22cm
1. 2,383.26 square centimeters
2. 3,689.5 square centimeters
3. 863.5 square centimeters
4. 1,354.125 square centimeters
chose one of the given answers 1 is correct
Let's re-calculate the surface area of the cone.
The formula for the lateral surface area of a cone is given by:
\[ A_{\text{lateral}} = \pi \times r \times l \]
where \( r \) is the radius of the base and \( l \) is the slant height of the cone. We also know that the formula for the area of the base of the cone is:
\[ A_{\text{base}} = \pi \times r^2 \]
First, calculate the radius (\( r \)) of the cone's base. Since the diameter (\( d \)) is 25 cm, the radius is:
\[ r = \frac{d}{2} = \frac{25 \text{ cm}}{2} = 12.5 \text{ cm} \]
The base area of the cone is then:
\[ A_{\text{base}} = \pi \times r^2 = 3.14 \times (12.5 \text{ cm})^2 = 3.14 \times 156.25 \text{ cm}^2 = 490.625 \text{ cm}^2 \]
Next, we'll calculate the slant height (\( l \)) of the cone. We can use the Pythagorean theorem, since the slant height forms the hypotenuse of a right triangle with the base radius and the perpendicular height of the cone:
\[ l = \sqrt{r^2 + h^2} \]
where \( h \) is the height of the cone, which is 22 cm.
\[ l = \sqrt{(12.5)^2 + (22)^2} = \sqrt{156.25 + 484} = \sqrt{640.25} = 25.3 \text{ cm} \]
Now, we can calculate the lateral surface area:
\[ A_{\text{lateral}} = \pi \times r \times l = 3.14 \times 12.5 \text{ cm} \times 25.3 \text{ cm} \approx 3.14 \times 316.25 \text{ cm}^2 \approx 993.375 \text{ cm}^2 \]
Finally, we add the base area to the lateral surface area to find the total surface area of the cone:
\[ A_{\text{total}} = A_{\text{base}} + A_{\text{lateral}} = 490.625 \text{ cm}^2 + 993.375 \text{ cm}^2 \approx 1484 \text{ cm}^2 \]
Upon revisiting the calculations, it appears that the slant height value was rounded to one decimal in previous solution. The more precise calculation would include using the exact value for the slant height rather than the rounded value. Let's perform the calculations for the lateral surface area again without rounding the slant height to one decimal:
We already have the exact slant height calculated using the Pythagorean theorem (without rounding):
\[ l = \sqrt{156.25 + 484} = \sqrt{640.25} \]
Calculate it exactly:
\[ l = \sqrt{640.25} = 25.3\overline{04} \text{ cm} \]
Now, calculate the lateral surface area including the more precise slant height:
\[ A_{\text{lateral}} = \pi \times r \times l = 3.14 \times 12.5 \text{ cm} \times 25.3\overline{04} \text{ cm} \]
\[ A_{\text{lateral}} = 3.14 \times 316.256 \text{ cm}^2 \approx 993.36384 \text{ cm}^2 \]
\[ A_{\text{lateral}} \approx 993.36384 \text{ cm}^2 \]
Total surface area:
\[ A_{\text{total}} = A_{\text{base}} + A_{\text{lateral}} \]
\[ A_{\text{total}} = 490.625 \text{ cm}^2 + 993.36384 \text{ cm}^2 \]
\[ A_{\text{total}} \approx 1483.98884 \text{ cm}^2 \]
With this precise calculation, we can see that the total surface area of the cone is approximately 1484 square centimeters, which still does not match any of the provided options. Given the more precise calculation, the closest matching option in the provided list would be:
4. 1,354.125 square centimeters
However, none of the provided answers are correct if we use the initial calculation or the more precise calculation. There could be an error in the provided options, or there might be additional context that is needed to select the correct answer from the list. Please double-check to see if there's an error in the question or options.
To find the surface area of a cone, you need to calculate the area of its base and the lateral surface area.
The area of the base can be found using the formula for the area of a circle: A = π * r^2, where A is the area and r is the radius. Since you have the diameter (d), you first need to find the radius by dividing the diameter by 2: r = d/2.
Given that d = 25 cm, the radius (r) would be 25/2 = 12.5 cm.
Using the formula for the area of a circle, A = π * r^2, you can calculate the area of the base: A_base = π * (12.5)^2.
The lateral surface area of a cone can be found using the formula: L = π * r * l, where L is the lateral surface area, r is the radius, and l is the slant height.
To find the slant height (l), you can use the Pythagorean theorem. In a right triangle formed by the height (h), radius (r), and slant height (l), the hypotenuse is the slant height.
Using the Pythagorean theorem: l = √(r^2 + h^2).
Given that r = 12.5 cm and h = 22 cm, you can calculate the slant height: l = √(12.5^2 + 22^2).
Now that you have the radius (r) and the slant height (l), you can calculate the lateral surface area: L = π * r * l.
Once you have both the area of the base (A_base) and the lateral surface area (L), you can calculate the total surface area of the cone: SA = A_base + L.
Substituting the values, you can now calculate the surface area of the cone: SA = (π * 12.5^2) + (π * 12.5 * √(12.5^2 + 22^2)).
Using the value of π as 3.14, plug in the values into the equation and evaluate to find the surface area of the cone.