To determine the frequency of the new note and its percentage increase relative to the frequency of the A above middle C, let's follow these steps:
1. Identify Known Values:
- Frequency of A above middle C ([tex]\( f_A \)[/tex]) = 440 Hz
- Number of half-steps away ([tex]\( h \)[/tex]) = 6
2. Determine Frequency of the New Note:
The formula for the frequency [tex]\( f \)[/tex] of a note that is [tex]\( h \)[/tex] half-steps away from A above middle C is:
[tex]\[
f = 440 \times 2^{\frac{h}{12}}
\][/tex]
Plugging in [tex]\( h = 6 \)[/tex]:
[tex]\[
f = 440 \times 2^{\frac{6}{12}}
\][/tex]
[tex]\[
f = 440 \times 2^{0.5}
\][/tex]
[tex]\[
f = 440 \times \sqrt{2}
\][/tex]
[tex]\[
f \approx 440 \times 1.41421356
\][/tex]
[tex]\[
f \approx 622.254 \, \text{Hz}
\][/tex]
3. Calculate the Percentage Increase:
The percentage increase in frequency is given by:
[tex]\[
\text{Percentage Increase} = \left( \frac{\text{New Frequency} - \text{Original Frequency}}{\text{Original Frequency}} \right) \times 100
\][/tex]
Using the values we have:
[tex]\[
\text{Percentage Increase} = \left( \frac{622.254 - 440}{440} \right) \times 100
\][/tex]
[tex]\[
\text{Percentage Increase} = \left( \frac{182.254}{440} \right) \times 100
\][/tex]
[tex]\[
\text{Percentage Increase} \approx 0.41421356 \times 100
\][/tex]
[tex]\[
\text{Percentage Increase} \approx 41.421 \%
\][/tex]
4. Find the Closest Choice:
Comparing the calculated percentage increase (41.421%) to the provided options:
- 29.3%
- 41.4%
- 70.7%
- 182.3%
The closest match to 41.421% is 41.4%.
So, the frequency of the new note is approximately 622.254 Hz, and the percentage increase in frequency compared to the A above middle C is approximately 41.4%.
Therefore, the correct answer is 41.4%.
