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4. The temperature in degrees Fahrenheit, [tex]F[/tex], is related to the temperature in degrees Celsius, [tex]C[/tex], by the equation:

[tex]F = \frac{9}{5}C + 32[/tex]

a. In the Sahara Desert, temperatures often reach 50 degrees Celsius. How many degrees Fahrenheit is this?

b. In parts of Alaska, the temperatures can reach -60 degrees Fahrenheit. How many degrees Celsius is this?

c. There is one temperature where the degrees Fahrenheit and degrees Celsius are the same, so that [tex]C = F[/tex]. Use the equation, where [tex]F[/tex] is expressed in terms of [tex]C[/tex], to solve for this temperature.

Answer :

Sure, let's go through the problem step by step.

### a. Convert 50 degrees Celsius to Fahrenheit

To convert a temperature from Celsius to Fahrenheit, we use the formula:
[tex]\[ F = \frac{9}{5}C + 32 \][/tex]

Here, [tex]\( C = 50 \)[/tex].

Substituting 50 into the formula:
[tex]\[ F = \frac{9}{5} \times 50 + 32 \][/tex]

First, multiply [tex]\( \frac{9}{5} \)[/tex] by 50:
[tex]\[ F = 90 + 32 \][/tex]

Finally, add 32:
[tex]\[ F = 122 \][/tex]

So, 50 degrees Celsius is 122 degrees Fahrenheit.

### b. Convert -60 degrees Fahrenheit to Celsius

To convert from Fahrenheit to Celsius, we use the formula:
[tex]\[ C = \frac{5}{9}(F - 32) \][/tex]

Here, [tex]\( F = -60 \)[/tex].

Substituting -60 into the formula:
[tex]\[ C = \frac{5}{9}(-60 - 32) \][/tex]

Calculate [tex]\(-60 - 32\)[/tex]:
[tex]\[ C = \frac{5}{9}(-92) \][/tex]

Finally, multiply [tex]\( \frac{5}{9} \)[/tex] by -92:
[tex]\[ C \approx -51.11 \][/tex]

So, -60 degrees Fahrenheit is approximately -51.11 degrees Celsius.

### c. Find the temperature where degrees Fahrenheit and degrees Celsius are the same

We know the equation relating Fahrenheit and Celsius is:
[tex]\[ F = \frac{9}{5}C + 32 \][/tex]

To find the temperature where they are the same, set [tex]\( F = C \)[/tex]:
[tex]\[ C = \frac{9}{5}C + 32 \][/tex]

Rearranging this:
[tex]\[ C - \frac{9}{5}C = 32 \][/tex]

Factor the left side:
[tex]\[ \left(1 - \frac{9}{5}\right)C = 32 \][/tex]

Calculate the coefficient of [tex]\( C \)[/tex]:
[tex]\[ \left(-\frac{4}{5}\right)C = 32 \][/tex]

Now solve for [tex]\( C \)[/tex]:
[tex]\[ C = \frac{32 \times 5}{-4} \][/tex]
[tex]\[ C = -40 \][/tex]

Therefore, the temperature at which Celsius and Fahrenheit are the same is -40 degrees.

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