To analyze the relationship between amplitude and energy and to find the missing data point, let's break down the problem step-by-step:
1. Given Data:
- Amplitudes: 6, 7, 8, 9, 10 units
- Corresponding energies: 72, 98, 128, ?, 200 units
2. Understanding the Relationship:
- We assume there is a relationship between amplitude (A) and energy (E). A common assumption is that energy is proportional to the square of the amplitude:
[tex]\[ E \propto A^2 \][/tex]
This means:
[tex]\[ E = k \cdot A^2 \][/tex]
where [tex]\( k \)[/tex] is a proportionality constant.
3. Calculating the Proportionality Constant [tex]\( k \)[/tex]:
- Using the known data points:
For [tex]\( A = 6 \)[/tex]:
[tex]\[ 72 = k \cdot 6^2 \][/tex]
[tex]\[ 72 = k \cdot 36 \][/tex]
[tex]\[ k = \frac{72}{36} = 2 \][/tex]
For [tex]\( A = 7 \)[/tex]:
[tex]\[ 98 = k \cdot 7^2 \][/tex]
[tex]\[ 98 = k \cdot 49 \][/tex]
[tex]\[ k = \frac{98}{49} = 2 \][/tex]
For [tex]\( A = 8 \)[/tex]:
[tex]\[ 128 = k \cdot 8^2 \][/tex]
[tex]\[ 128 = k \cdot 64 \][/tex]
[tex]\[ k = \frac{128}{64} = 2 \][/tex]
For [tex]\( A = 10 \)[/tex]:
[tex]\[ 200 = k \cdot 10^2 \][/tex]
[tex]\[ 200 = k \cdot 100 \][/tex]
[tex]\[ k = \frac{200}{100} = 2 \][/tex]
- By examining the data, we observe that the constant [tex]\( k \)[/tex] is consistently 2 across all provided data points.
4. Calculating the Missing Energy:
- For [tex]\( A = 9 \)[/tex]:
[tex]\[ E = k \cdot 9^2 \][/tex]
[tex]\[ E = 2 \cdot 81 \][/tex]
[tex]\[ E = 162 \][/tex]
5. Identifying the Answer Among the Options:
- The calculated missing energy is 162 units.
6. Conclusion:
- The missing energy for an amplitude of 9 units is 162 units.
Therefore, the correct answer is:
C. 162
