Find the area of the iron sheet required to prepare a cone (with base) is 24 cm height with base radius 7 cm :

Question

Accepted Answer

To find the area of the iron sheet required to prepare a cone with a base, we need to calculate the total surface area of the cone. The total surface area of a cone with a base is the sum of the lateral surface area and the area of the base.

Given:
- Height of the cone (h) = 24 cm
- Radius of the base (r) = 7 cm

First, we need to find the slant height (l) of the cone using the Pythagorean theorem:
$$ l = \sqrt{r^2 + h^2} $$
$$ l = \sqrt{7^2 + 24^2} $$
$$ l = \sqrt{49 + 576} $$
$$ l = \sqrt{625} $$
$$ l = 25 \, \text{cm} $$

Now, calculate the lateral surface area of the cone:
$$ \text{Lateral Surface Area} = \pi \times r \times l $$
$$ \text{Lateral Surface Area} = \pi \times 7 \times 25 $$
$$ \text{Lateral Surface Area} = 175\pi $$

Next, calculate the area of the base of the cone:
$$ \text{Base Area} = \pi \times r^2 $$
$$ \text{Base Area} = \pi \times 7^2 $$
$$ \text{Base Area} = 49\pi $$

The total surface area of the cone is the sum of the lateral surface area and the base area:
$$ \text{Total Surface Area} = 175\pi + 49\pi $$
$$ \text{Total Surface Area} = 224\pi $$

Using the approximation $\pi \approx 3.14$, we calculate:
$$ \text{Total Surface Area} = 224 \times 3.14 $$
$$ \text{Total Surface Area} \approx 703.36 \, \text{cm}^2 $$

Rounding to the nearest whole number, the total surface area is approximately 704 cm².

Therefore, the correct answer is Option 3: 704 cm².

Explanation for other options:
1. **625 cm²**: This option is incorrect because it does not account for the correct calculation of the total surface area using the given dimensions.
2. **690 cm²**: This option is incorrect as it underestimates the total surface area calculated using the formula and given dimensions.
4. **729 cm²**: This option is incorrect because it overestimates the total surface area calculated using the formula and given dimensions.

Thus, the correct answer is indeed Option 3: 704 cm².