Find the value of question mark (?) in the following questions.
20% of `(40 xx sqrt(?)) = (32)^(2) + (16)^(2)`

Options

A. 160
B. 2560
C. 16
D. 25600

Answer

To find the value of the question mark (?) in the equation:

\[ 20\% \text{ of } (40 \times \sqrt{?}) = (32)^2 + (16)^2 \]

we will solve it step by step.

### Step 1: Convert the percentage to a decimal
First, we convert 20% into a decimal for easier calculations.

\[ 20\% = \frac{20}{100} = 0.2 \]

### Step 2: Rewrite the equation
Now, we can rewrite the equation using the decimal:

\[ 0.2 \times (40 \times \sqrt{?}) = (32)^2 + (16)^2 \]

### Step 3: Calculate the right side of the equation
Next, we calculate \( (32)^2 \) and \( (16)^2 \):

\[ (32)^2 = 1024 \]
\[ (16)^2 = 256 \]

Now, add these two results:

\[ 1024 + 256 = 1280 \]

### Step 4: Substitute back into the equation
Now, substitute this back into our equation:

\[ 0.2 \times (40 \times \sqrt{?}) = 1280 \]

### Step 5: Isolate the term with the square root
To isolate \( 40 \times \sqrt{?} \), divide both sides by 0.2:

\[ 40 \times \sqrt{?} = \frac{1280}{0.2} \]

Calculating the right side:

\[ \frac{1280}{0.2} = 1280 \times 5 = 6400 \]

### Step 6: Solve for \( \sqrt{?} \)
Now we have:

\[ 40 \times \sqrt{?} = 6400 \]

Next, divide both sides by 40:

\[ \sqrt{?} = \frac{6400}{40} \]

Calculating the right side:

\[ \frac{6400}{40} = 160 \]

### Step 7: Square both sides to find ?
Now, square both sides to solve for ?:

\[ ? = (160)^2 \]

Calculating this gives:

\[ ? = 25600 \]

### Final Answer
Thus, the value of the question mark (?) is:

\[ \boxed{25600} \]

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