Answer :
Sure, let's solve the problem step-by-step:
### Part (a): Finding the Inverse of the Formula
The original formula provided for converting Celsius to Fahrenheit is:
[tex]\[ F = \frac{9}{5}C + 32 \][/tex]
To find the inverse, we'll solve for [tex]\( C \)[/tex] in terms of [tex]\( F \)[/tex]. Here's how you can do it:
1. Start by subtracting 32 from both sides of the equation to isolate the term with [tex]\( C \)[/tex]:
[tex]\[ F - 32 = \frac{9}{5}C \][/tex]
2. Next, multiply both sides by the reciprocal of [tex]\(\frac{9}{5}\)[/tex], which is [tex]\(\frac{5}{9}\)[/tex], to solve for [tex]\( C \)[/tex]:
[tex]\[ C = \frac{5}{9}(F - 32) \][/tex]
Now we have the inverse formula:
[tex]\[ C = \frac{5}{9}(F - 32) \][/tex]
This inverse is indeed a function because for every input Fahrenheit value [tex]\( F \)[/tex], there is a unique output Celsius value [tex]\( C \)[/tex].
### Part (b): Finding the Celsius Temperature Corresponding to [tex]\( 16^\circ F \)[/tex]
To find the Celsius temperature for [tex]\( 16^\circ F \)[/tex], use the inverse formula we just derived:
[tex]\[ C = \frac{5}{9}(F - 32) \][/tex]
Substitute [tex]\( F = 16 \)[/tex] into the formula:
[tex]\[ C = \frac{5}{9}(16 - 32) \][/tex]
[tex]\[ C = \frac{5}{9}(-16) \][/tex]
[tex]\[ C = -\frac{80}{9} \][/tex]
[tex]\[ C \approx -8.89 \][/tex]
So, the Celsius temperature that corresponds to [tex]\( 16^\circ F \)[/tex] is approximately [tex]\( -8.89^\circ C \)[/tex].
### Part (a): Finding the Inverse of the Formula
The original formula provided for converting Celsius to Fahrenheit is:
[tex]\[ F = \frac{9}{5}C + 32 \][/tex]
To find the inverse, we'll solve for [tex]\( C \)[/tex] in terms of [tex]\( F \)[/tex]. Here's how you can do it:
1. Start by subtracting 32 from both sides of the equation to isolate the term with [tex]\( C \)[/tex]:
[tex]\[ F - 32 = \frac{9}{5}C \][/tex]
2. Next, multiply both sides by the reciprocal of [tex]\(\frac{9}{5}\)[/tex], which is [tex]\(\frac{5}{9}\)[/tex], to solve for [tex]\( C \)[/tex]:
[tex]\[ C = \frac{5}{9}(F - 32) \][/tex]
Now we have the inverse formula:
[tex]\[ C = \frac{5}{9}(F - 32) \][/tex]
This inverse is indeed a function because for every input Fahrenheit value [tex]\( F \)[/tex], there is a unique output Celsius value [tex]\( C \)[/tex].
### Part (b): Finding the Celsius Temperature Corresponding to [tex]\( 16^\circ F \)[/tex]
To find the Celsius temperature for [tex]\( 16^\circ F \)[/tex], use the inverse formula we just derived:
[tex]\[ C = \frac{5}{9}(F - 32) \][/tex]
Substitute [tex]\( F = 16 \)[/tex] into the formula:
[tex]\[ C = \frac{5}{9}(16 - 32) \][/tex]
[tex]\[ C = \frac{5}{9}(-16) \][/tex]
[tex]\[ C = -\frac{80}{9} \][/tex]
[tex]\[ C \approx -8.89 \][/tex]
So, the Celsius temperature that corresponds to [tex]\( 16^\circ F \)[/tex] is approximately [tex]\( -8.89^\circ C \)[/tex].
