A toy is in the form of a cylinder with hemispherical ends. If the whole length of the toy is 90 cm and its diameter is 42 cm , find the cost of painting the toy at the rate of 70 paise per sq m .

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the dimensions of the toy
The toy is in the shape of a cylinder with two hemispherical ends. We know:
- Total length of the toy = 90 cm
- Diameter of the toy = 42 cm

From the diameter, we can find the radius:
- Radius (r) = Diameter / 2 = 42 cm / 2 = 21 cm

### Step 2: Calculate the height of the cylindrical part
Since the toy has two hemispherical ends, the total length of the toy includes the height of the cylinder and the diameters of the two hemispheres. The height of the two hemispheres combined is equal to the diameter of the toy:
- Height of the two hemispheres = Diameter = 42 cm

Now, we can find the height of the cylindrical part (H):
- Total length = Height of cylinder + Height of two hemispheres
- 90 cm = H + 42 cm
- H = 90 cm - 42 cm = 48 cm

### Step 3: Calculate the curved surface area (CSA) of the cylinder
The formula for the curved surface area of a cylinder is given by:
\[ \text{CSA}_{\text{cylinder}} = 2\pi rh \]
Where:
- r = radius of the cylinder = 21 cm
- h = height of the cylinder = 48 cm

Substituting the values:
\[ \text{CSA}_{\text{cylinder}} = 2 \times \frac{22}{7} \times 21 \times 48 \]

Calculating:
\[ \text{CSA}_{\text{cylinder}} = 2 \times \frac{22}{7} \times 21 \times 48 = 2 \times \frac{22 \times 21 \times 48}{7} \]
\[ = 2 \times \frac{22032}{7} = \frac{44064}{7} \approx 6309.14 \text{ cm}^2 \]

### Step 4: Calculate the surface area of the two hemispherical ends
The formula for the surface area of a sphere is:
\[ \text{SA}_{\text{sphere}} = 4\pi r^2 \]
Since we have two hemispheres, the total surface area of the two hemispheres is:
\[ \text{SA}_{\text{hemispheres}} = 2 \times \text{SA}_{\text{sphere}} = 2 \times 4\pi r^2 = 8\pi r^2 \]

Substituting the radius:
\[ \text{SA}_{\text{hemispheres}} = 8 \times \frac{22}{7} \times (21)^2 \]
\[ = 8 \times \frac{22}{7} \times 441 = 8 \times \frac{9702}{7} = \frac{77616}{7} \approx 11023.71 \text{ cm}^2 \]

### Step 5: Calculate the total surface area of the toy
Now, we can find the total surface area (TSA) of the toy:
\[ \text{TSA} = \text{CSA}_{\text{cylinder}} + \text{SA}_{\text{hemispheres}} \]
\[ = 6309.14 + 11023.71 \approx 17332.85 \text{ cm}^2 \]

### Step 6: Convert the total surface area into square meters
Since the cost is given per square meter, we need to convert square centimeters to square meters:
\[ 1 \text{ m}^2 = 10000 \text{ cm}^2 \]
So,
\[ \text{TSA in m}^2 = \frac{17332.85}{10000} \approx 1.733285 \text{ m}^2 \]

### Step 7: Calculate the cost of painting the toy
The cost of painting is given as 70 paise per square meter. First, convert paise to rupees:
\[ 70 \text{ paise} = 0.70 \text{ rupees} \]

Now, calculate the total cost:
\[ \text{Cost} = \text{TSA in m}^2 \times \text{Cost per m}^2 \]
\[ = 1.733285 \times 0.70 \approx 1.2133 \text{ rupees} \]

### Final Answer
The cost of painting the toy is approximately **₹1.21**.