Neziswa has just bought a new radio. She wants to be able to tune into her favourite radio station, Ukhozi FM. This radio station broadcasts over a frequency range of $$ 90,8 \mathrm{MHz}-93,4 \mathrm{MHz} $$. Neziswa reads the manual for her radio and finds that her radio can receive signals that have a wavelength oetween $$ 2,78 \mathrm{~m} $$ and $$ 3,8 \mathrm{~m} $$.

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To determine if Neziswa's radio can tune into Ukhozi FM, we need to convert the frequency range of the radio station into wavelengths and see if those wavelengths fall within the range that her radio can receive.

### Step 1: Understand the relationship between frequency and wavelength

The relationship between frequency ($$ f $$) and wavelength ($$ \lambda $$) is given by the formula:

$$
\lambda = \frac{c}{f}
$$

where:
- $$ \lambda $$ is the wavelength in meters,
- $$ c $$ is the speed of light in a vacuum, approximately $$ 3 \times 10^8 $$ m/s,
- $$ f $$ is the frequency in hertz (Hz).

### Step 2: Convert the frequency range from MHz to Hz

The frequency range for Ukhozi FM is given as $$ 90.8 $$ MHz to $$ 93.4 $$ MHz. We need to convert these values to hertz:

$$
90.8 \, \text{MHz} = 90.8 \times 10^6 \, \text{Hz}
$$
$$
93.4 \, \text{MHz} = 93.4 \times 10^6 \, \text{Hz}
$$

### Step 3: Calculate the corresponding wavelengths

Now we will calculate the wavelengths for both frequencies using the formula mentioned above.

1. For $$ 90.8 \, \text{MHz} $$:
   $$
   \lambda_1 = \frac{c}{f_1} = \frac{3 \times 10^8}{90.8 \times 10^6}
   $$

2. For $$ 93.4 \, \text{MHz} $$:
   $$
   \lambda_2 = \frac{c}{f_2} = \frac{3 \times 10^8}{93.4 \times 10^6}
   $$

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{3\times 10^{8}}{\left(93.4\times 10^{6}\right)}$$
- step1: Remove the parentheses:
  $$\frac{3\times 10^{8}}{93.4\times 10^{6}}$$
- step2: Factor the expression:
  $$\frac{10^{6}\times 300}{93.4\times 10^{6}}$$
- step3: Reduce the fraction:
  $$\frac{300}{93.4}$$
- step4: Convert the expressions:
  $$\frac{300}{\frac{467}{5}}$$
- step5: Multiply by the reciprocal:
  $$300\times \frac{5}{467}$$
- step6: Multiply:
  $$\frac{300\times 5}{467}$$
- step7: Multiply:
  $$\frac{1500}{467}$$
Calculate or simplify the expression $$ 3 * 10^8 / (90.8 * 10^6) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{3\times 10^{8}}{\left(90.8\times 10^{6}\right)}$$
- step1: Remove the parentheses:
  $$\frac{3\times 10^{8}}{90.8\times 10^{6}}$$
- step2: Factor the expression:
  $$\frac{10^{6}\times 300}{90.8\times 10^{6}}$$
- step3: Reduce the fraction:
  $$\frac{300}{90.8}$$
- step4: Convert the expressions:
  $$\frac{300}{\frac{454}{5}}$$
- step5: Multiply by the reciprocal:
  $$300\times \frac{5}{454}$$
- step6: Reduce the numbers:
  $$150\times \frac{5}{227}$$
- step7: Multiply:
  $$\frac{150\times 5}{227}$$
- step8: Multiply:
  $$\frac{750}{227}$$
### Step 4: Calculate the wavelengths

1. For $$ 90.8 \, \text{MHz} $$:
   $$
   \lambda_1 \approx 3.304 \, \text{m}
   $$

2. For $$ 93.4 \, \text{MHz} $$:
   $$
   \lambda_2 \approx 3.212 \, \text{m}
   $$

### Step 5: Determine the wavelength range for Ukhozi FM

The wavelengths corresponding to the frequency range of Ukhozi FM are approximately:
- Minimum wavelength: $$ \lambda_2 \approx 3.212 \, \text{m} $$
- Maximum wavelength: $$ \lambda_1 \approx 3.304 \, \text{m} $$

### Step 6: Compare with Neziswa's radio wavelength range

Neziswa's radio can receive signals with wavelengths between $$ 2.78 \, \text{m} $$ and $$ 3.8 \, \text{m} $$.

### Step 7: Check if the ranges overlap

- Ukhozi FM wavelength range: $$ 3.212 \, \text{m} $$ to $$ 3.304 \, \text{m} $$
- Neziswa's radio wavelength range: $$ 2.78 \, \text{m} $$ to $$ 3.8 \, \text{m} $$

Since $$ 3.212 \, \text{m} $$ to $$ 3.304 \, \text{m} $$ falls within $$ 2.78 \, \text{m} $$ to $$ 3.8 \, \text{m} $$, we can conclude that:

### Final Conclusion

Neziswa's radio can indeed tune into her favorite radio station, Ukhozi FM, as the wavelength range of the station falls within the receiving capabilities of her radio.
Neziswa's radio can tune into Ukhozi FM because the station's wavelength range (approximately 3.212 m to 3.304 m) falls within her radio's reception range of 2.78 m to 3.8 m.

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