The value of : `[(83.7 xx 83.7 + 16.3 xx 16.3 - 83.7 xx 16.3)/(0.837 xx 0.837 xx 0.837 + 0.163 xx 0.163 xx 0.163)] div 333(1)/(3)`

Options

A. 25
B. 30
C. 33
D. 0.3

Answer

To solve the expression:

\[ \frac{(83.7 \times 83.7 + 16.3 \times 16.3 - 83.7 \times 16.3)}{(0.837 \times 0.837 \times 0.837 + 0.163 \times 0.163 \times 0.163)} \div 333\frac{1}{3} \]

we will break it down step by step.

### Step 1: Simplify the Numerator
The numerator is:

\[ 83.7^2 + 16.3^2 - 83.7 \times 16.3 \]

Calculating each term:

- \(83.7^2 = 7009.69\)
- \(16.3^2 = 265.69\)
- \(83.7 \times 16.3 = 1367.51\)

Now substitute these values back into the numerator:

\[ 7009.69 + 265.69 - 1367.51 \]

Calculating this gives:

\[ 7009.69 + 265.69 = 7275.38 \]
\[ 7275.38 - 1367.51 = 5907.87 \]

So, the numerator simplifies to:

\[ 5907.87 \]

### Step 2: Simplify the Denominator
The denominator is:

\[ 0.837^3 + 0.163^3 \]

Calculating each term:

- \(0.837^3 = 0.586\) (approximately)
- \(0.163^3 = 0.0044\) (approximately)

Now substitute these values back into the denominator:

\[ 0.586 + 0.0044 = 0.5904 \]

### Step 3: Combine Numerator and Denominator
Now we can combine the results from Steps 1 and 2:

\[ \frac{5907.87}{0.5904} \]

Calculating this gives:

\[ 5907.87 \div 0.5904 \approx 10000 \]

### Step 4: Divide by \(333\frac{1}{3}\)
Now we need to divide by \(333\frac{1}{3}\), which is:

\[ 333\frac{1}{3} = \frac{1000}{3} \]

So we need to perform the division:

\[ 10000 \div \frac{1000}{3} = 10000 \times \frac{3}{1000} = 30 \]

### Final Answer
Thus, the final value of the expression is:

\[ \boxed{30} \]