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Sagot :
To determine which transformer Roberto should use to achieve an ending voltage of [tex]\( 4 \, \text{V} \)[/tex], let's go through the problem step by step.
### Step 1: First Transformer Calculation
Roberto uses the first transformer with:
- Primary winding: 300 coils
- Secondary winding: 50 coils
He starts with an initial voltage of [tex]\( 120 \, \text{V} \)[/tex].
Given the formula for the voltage transformation:
[tex]\[ V_s = V_p \times \frac{N_s}{N_p} \][/tex]
where:
- [tex]\( V_s \)[/tex] is the secondary voltage.
- [tex]\( V_p \)[/tex] is the primary voltage.
- [tex]\( N_s \)[/tex] is the number of secondary coils.
- [tex]\( N_p \)[/tex] is the number of primary coils.
Substitute the given values:
[tex]\[ V_s = 120 \, \text{V} \times \frac{50}{300} \][/tex]
Calculating the fraction:
[tex]\[ V_s = 120 \, \text{V} \times \frac{1}{6} = 20 \, \text{V} \][/tex]
The intermediate voltage after the first transformer is [tex]\( 20 \, \text{V} \)[/tex].
### Step 2: Second Transformer Calculation
Roberto now needs to reduce the [tex]\( 20 \, \text{V} \)[/tex] intermediate voltage down to [tex]\( 4 \, \text{V} \)[/tex] using one of the available transformers. Let's check each transformer to find the right one:
#### Transformer W:
- Primary winding: 80 coils
- Secondary winding: 20 coils
The voltage transformation formula:
[tex]\[ V_s = V_p \times \frac{N_s}{N_p} \][/tex]
[tex]\[ V_s = 20 \, \text{V} \times \frac{20}{80} = 20 \, \text{V} \times \frac{1}{4} = 5 \, \text{V} \][/tex]
This does not give us the desired [tex]\( 4 \, \text{V} \)[/tex].
#### Transformer X:
- Primary winding: 60 coils
- Secondary winding: 12 coils
Using the formula:
[tex]\[ V_s = V_p \times \frac{N_s}{N_p} \][/tex]
[tex]\[ V_s = 20 \, \text{V} \times \frac{12}{60} = 20 \, \text{V} \times \frac{1}{5} = 4 \, \text{V} \][/tex]
This gives us the exact desired [tex]\( 4 \, \text{V} \)[/tex].
#### Transformer Y:
- Primary winding: 70 coils
- Secondary winding: 35 coils
Using the formula:
[tex]\[ V_s = V_p \times \frac{N_s}{N_p} \][/tex]
[tex]\[ V_s = 20 \, \text{V} \times \frac{35}{70} = 20 \, \text{V} \times \frac{1}{2} = 10 \, \text{V} \][/tex]
This does not give us the desired [tex]\( 4 \, \text{V} \)[/tex].
#### Transformer Z:
- Primary winding: 50 coils
- Secondary winding: 5 coils
Using the formula:
[tex]\[ V_s = V_p \times \frac{N_s}{N_p} \][/tex]
[tex]\[ V_s = 20 \, \text{V} \times \frac{5}{50} = 20 \, \text{V} \times \frac{1}{10} = 2 \, \text{V} \][/tex]
This does not give us the desired [tex]\( 4 \, \text{V} \)[/tex].
### Conclusion
From the calculations, we see that Transformer X is the one that achieves the ending voltage of [tex]\( 4 \, \text{V} \)[/tex]. Therefore, Roberto should use transformer [tex]\( \boxed{X} \)[/tex].
### Step 1: First Transformer Calculation
Roberto uses the first transformer with:
- Primary winding: 300 coils
- Secondary winding: 50 coils
He starts with an initial voltage of [tex]\( 120 \, \text{V} \)[/tex].
Given the formula for the voltage transformation:
[tex]\[ V_s = V_p \times \frac{N_s}{N_p} \][/tex]
where:
- [tex]\( V_s \)[/tex] is the secondary voltage.
- [tex]\( V_p \)[/tex] is the primary voltage.
- [tex]\( N_s \)[/tex] is the number of secondary coils.
- [tex]\( N_p \)[/tex] is the number of primary coils.
Substitute the given values:
[tex]\[ V_s = 120 \, \text{V} \times \frac{50}{300} \][/tex]
Calculating the fraction:
[tex]\[ V_s = 120 \, \text{V} \times \frac{1}{6} = 20 \, \text{V} \][/tex]
The intermediate voltage after the first transformer is [tex]\( 20 \, \text{V} \)[/tex].
### Step 2: Second Transformer Calculation
Roberto now needs to reduce the [tex]\( 20 \, \text{V} \)[/tex] intermediate voltage down to [tex]\( 4 \, \text{V} \)[/tex] using one of the available transformers. Let's check each transformer to find the right one:
#### Transformer W:
- Primary winding: 80 coils
- Secondary winding: 20 coils
The voltage transformation formula:
[tex]\[ V_s = V_p \times \frac{N_s}{N_p} \][/tex]
[tex]\[ V_s = 20 \, \text{V} \times \frac{20}{80} = 20 \, \text{V} \times \frac{1}{4} = 5 \, \text{V} \][/tex]
This does not give us the desired [tex]\( 4 \, \text{V} \)[/tex].
#### Transformer X:
- Primary winding: 60 coils
- Secondary winding: 12 coils
Using the formula:
[tex]\[ V_s = V_p \times \frac{N_s}{N_p} \][/tex]
[tex]\[ V_s = 20 \, \text{V} \times \frac{12}{60} = 20 \, \text{V} \times \frac{1}{5} = 4 \, \text{V} \][/tex]
This gives us the exact desired [tex]\( 4 \, \text{V} \)[/tex].
#### Transformer Y:
- Primary winding: 70 coils
- Secondary winding: 35 coils
Using the formula:
[tex]\[ V_s = V_p \times \frac{N_s}{N_p} \][/tex]
[tex]\[ V_s = 20 \, \text{V} \times \frac{35}{70} = 20 \, \text{V} \times \frac{1}{2} = 10 \, \text{V} \][/tex]
This does not give us the desired [tex]\( 4 \, \text{V} \)[/tex].
#### Transformer Z:
- Primary winding: 50 coils
- Secondary winding: 5 coils
Using the formula:
[tex]\[ V_s = V_p \times \frac{N_s}{N_p} \][/tex]
[tex]\[ V_s = 20 \, \text{V} \times \frac{5}{50} = 20 \, \text{V} \times \frac{1}{10} = 2 \, \text{V} \][/tex]
This does not give us the desired [tex]\( 4 \, \text{V} \)[/tex].
### Conclusion
From the calculations, we see that Transformer X is the one that achieves the ending voltage of [tex]\( 4 \, \text{V} \)[/tex]. Therefore, Roberto should use transformer [tex]\( \boxed{X} \)[/tex].
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