Length of a mathematics lab is `1(1)/(3)` of its breadth and its height is `(1)/(2)` of its length . The cost of whitewashing the walls at the rate of Rs. `2.60` per `m^(2)` is Rs. `291.20` . Find the cost of tiling the floor at the rate of Rs. `6.75` per `m^(2)` .

Options

A. Rs. `324`
B. Rs. `624`
C. Rs. `570`
D. Rs. `420`

Answer

To solve the problem step-by-step, let's break it down:

### Step 1: Define the Variables
Let the breadth of the mathematics lab be \( b \) meters. According to the problem:
- The length \( l \) is given as \( l = \frac{4}{3}b \).
- The height \( h \) is given as \( h = \frac{1}{2}l = \frac{1}{2} \times \frac{4}{3}b = \frac{2}{3}b \).

### Step 2: Calculate the Surface Area of the Walls
The surface area \( A \) of the walls of a cuboid is given by the formula:
\[ A = 2(lh + bh) \]
Substituting the values of \( l \) and \( h \):
\[ A = 2\left(\left(\frac{4}{3}b\right)\left(\frac{2}{3}b\right) + b\left(\frac{2}{3}b\right)\right) \]

Calculating each term:
1. \( lh = \frac{4}{3}b \times \frac{2}{3}b = \frac{8}{9}b^2 \)
2. \( bh = b \times \frac{2}{3}b = \frac{2}{3}b^2 \)

Now substituting back into the area formula:
\[ A = 2\left(\frac{8}{9}b^2 + \frac{2}{3}b^2\right) \]
To add the fractions, convert \( \frac{2}{3}b^2 \) to have a common denominator:
\[ \frac{2}{3}b^2 = \frac{6}{9}b^2 \]
So,
\[ A = 2\left(\frac{8}{9}b^2 + \frac{6}{9}b^2\right) = 2\left(\frac{14}{9}b^2\right) = \frac{28}{9}b^2 \]

### Step 3: Calculate the Cost of Whitewashing the Walls
The cost of whitewashing is given as Rs. 2.60 per \( m^2 \) and the total cost is Rs. 291.20. Therefore, we can set up the equation:
\[ \text{Cost} = \text{Area} \times \text{Rate} \]
\[ 291.20 = \frac{28}{9}b^2 \times 2.60 \]
To find \( b^2 \), rearranging gives:
\[ b^2 = \frac{291.20 \times 9}{28 \times 2.60} \]

Calculating the right-hand side:
\[ b^2 = \frac{2912}{72.8} = 40 \]
Taking the square root gives:
\[ b = \sqrt{40} = 6.32 \text{ (approximately)} \]

### Step 4: Calculate the Length
Using the value of \( b \):
\[ l = \frac{4}{3}b = \frac{4}{3} \times 6.32 \approx 8.43 \text{ (approximately)} \]

### Step 5: Calculate the Area of the Floor
The area of the floor \( A_f \) is given by:
\[ A_f = l \times b = 8.43 \times 6.32 \approx 53.34 \text{ (approximately)} \]

### Step 6: Calculate the Cost of Tiling the Floor
The cost of tiling is Rs. 6.75 per \( m^2 \):
\[ \text{Cost of tiling} = A_f \times 6.75 \approx 53.34 \times 6.75 \approx 360.00 \text{ (approximately)} \]

### Final Answer
The cost of tiling the floor is approximately Rs. 360.00.

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