The Pay-It-Forward Foundation received 1,024 applications for the 4 business initiative grants that it is awarding this year. The applications will go through several rounds of evaluation based on established criteria and scored accordingly. After each round of evaluation, half of the applications will be eliminated from consideration for one of this year's business initiative grants.

Question

Drag the appropriate value to each box to complete the equation to determine after how many rounds only 4 applications will remain.

Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.

___=___(__)^x

Answer 1

To determine how many rounds of evaluation it takes for only 4 applications to remain from an initial 1,024 applications, we can use the equation:

\[ 1024 = 4 \times (1/2)^x \]

We can rearrange this into a more standard form:

\[ 1024 = 4 \cdot \left(\frac{1}{2}\right)^x \]

This means we can rewrite it as:

\[ 1024 = 4 \cdot \left(2^{-1}\right)^x \]

Since \( (2^{-1})^x = 2^{-x} \), it follows that:

\[ 1024 = 4 \cdot 2^{-x} \]

Now, expressing everything in terms of powers of 2:

• We know that \( 1024 = 2^{10} \)
• And \( 4 = 2^2 \)

Thus, the equation becomes:

\[ 2^{10} = 2^2 \cdot 2^{-x} \]

Which simplifies to:

\[ 2^{10} = 2^{2 - x} \]

Since the bases are the same, we can equate the exponents:

\[ 10 = 2 - x \]

Solving for \(x\):

\[ x = 2 - 10 = -8 \]

However, it seems like we've gone slightly off. We instead want \(2^x\) instead of \(-x\). So ultimately we realize that calculation of x should yield rounds until 4 is reached:

If starting from 1024, we can divide it successively until 4 remains.

1. After 1 round: 1024 / 2 = 512
2. After 2 rounds: 512 / 2 = 256
3. After 3 rounds: 256 / 2 = 128
4. After 4 rounds: 128 / 2 = 64
5. After 5 rounds: 64 / 2 = 32
6. After 6 rounds: 32 / 2 = 16
7. After 7 rounds: 16 / 2 = 8
8. After 8 rounds: 8 / 2 = 4

Therefore, your answer should be

1024 = 4(1/2)^8

So the appropriate values to fill in the blanks are:

=(1/2)^8

Where the first blank is 1024, the second blank is 4, and x is 8.