Q.11. A hemispherical dome needs to be painted. If the circumference of the base of the dome is 154 cm , then find the cost of painting it if the cost of painting is $$ ₹ 4 $$ per $$ 100 \mathrm{~cm}^{2} $$ (use $$ \pi=\frac{22}{7} $$ ).

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To solve the problem, we need to follow these steps:

1. **Extract known conditions:**
   - Circumference of the base of the dome: $$ C = 154 \, \text{cm} $$
   - Cost of painting: $$ ₹ 4 $$ per $$ 100 \, \text{cm}^2 $$
   - Value of $$ \pi $$: $$ \pi = \frac{22}{7} $$

2. **Find the radius of the hemispherical dome:**
   The circumference of a circle is given by the formula:
   $$
   C = 2\pi r
   $$
   Rearranging this to find the radius $$ r $$:
   $$
   r = \frac{C}{2\pi}
   $$

3. **Calculate the surface area of the hemispherical dome:**
   The surface area $$ A $$ of a hemisphere (including the base) is given by:
   $$
   A = 2\pi r^2 + \pi r^2 = \frac{3}{2}\pi r^2
   $$
   However, since we only need to paint the curved surface area, we will use:
   $$
   A = 2\pi r^2
   $$

4. **Calculate the cost of painting:**
   The cost of painting is calculated based on the area to be painted.

Now, let's perform the calculations step by step.

### Step 1: Calculate the radius $$ r $$

Using the circumference:
$$
r = \frac{154}{2 \times \frac{22}{7}} = \frac{154 \times 7}{44} = \frac{1078}{44} = 24.5 \, \text{cm}
$$

### Step 2: Calculate the surface area $$ A $$

Now, substituting $$ r $$ into the surface area formula:
$$
A = 2\pi r^2 = 2 \times \frac{22}{7} \times (24.5)^2
$$

### Step 3: Calculate the cost of painting

The cost of painting is given by:
$$
\text{Cost} = \frac{A}{100} \times 4
$$

Now, let's perform the calculations.

First, we will calculate the area $$ A $$ and then the cost.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{A}{100}\times 4$$
- step1: Reduce the fraction:
  $$\frac{A}{25}\times 1$$
- step2: Multiply the terms:
  $$\frac{A}{25}$$
Calculate or simplify the expression $$ 2*(22/7)*(24.5^2) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2\times \frac{22}{7}\times 24.5^{2}$$
- step1: Convert the expressions:
  $$2\times \frac{22}{7}\left(\frac{49}{2}\right)^{2}$$
- step2: Multiply the terms:
  $$\frac{44}{7}\left(\frac{49}{2}\right)^{2}$$
- step3: Evaluate the power:
  $$\frac{44}{7}\times \frac{49^{2}}{2^{2}}$$
- step4: Rewrite the expression:
  $$\frac{4\times 11}{7}\times \frac{49^{2}}{2^{2}}$$
- step5: Rewrite the expression:
  $$\frac{2^{2}\times 11}{7}\times \frac{49^{2}}{2^{2}}$$
- step6: Reduce the numbers:
  $$\frac{11}{7}\times 49^{2}$$
- step7: Rewrite the expression:
  $$\frac{11}{7}\times 7^{4}$$
- step8: Reduce the numbers:
  $$11\times 7^{3}$$
- step9: Evaluate the power:
  $$11\times 343$$
- step10: Multiply:
  $$3773$$
Calculate or simplify the expression $$ (3773/100)*4 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{3773}{100}\times 4$$
- step1: Reduce the numbers:
  $$\frac{3773}{25}\times 1$$
- step2: Multiply:
  $$\frac{3773}{25}$$
### Step 1: Calculate the radius $$ r $$

The radius of the hemispherical dome is:
$$
r = 24.5 \, \text{cm}
$$

### Step 2: Calculate the surface area $$ A $$

The curved surface area of the hemispherical dome is:
$$
A = 2\pi r^2 = 2 \times \frac{22}{7} \times (24.5)^2 \approx 3773 \, \text{cm}^2
$$

### Step 3: Calculate the cost of painting

The cost of painting is calculated as follows:
$$
\text{Cost} = \frac{A}{100} \times 4 = \frac{3773}{25} \approx 150.92
$$

### Final Answer

The cost of painting the hemispherical dome is approximately $$ ₹ 150.92 $$.
The cost of painting the hemispherical dome is approximately ₹150.92.

Q.11. A hemispherical dome needs to be painted. If the circumference of the base of the dome is 154 cm , then find the cost of painting it if the cost of painting is \( ₹ 4 \) per \( 100 \mathrm{~cm}^{2} \) (use \( \pi=\frac{22}{7} \) ).

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