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Answer :
Certainly! Let's work through the problem step-by-step.
a. Finding the inverse of the formula:
The original formula for converting from Celsius to Fahrenheit is:
[tex]\[ F = \frac{9}{5} C + 32 \][/tex]
To find the inverse of this formula, we want to solve for [tex]\( C \)[/tex] in terms of [tex]\( F \)[/tex]. This involves isolating [tex]\( C \)[/tex] on one side of the equation.
1. Start with the equation:
[tex]\[ F = \frac{9}{5} C + 32 \][/tex]
2. To isolate [tex]\( C \)[/tex], first subtract 32 from both sides:
[tex]\[ F - 32 = \frac{9}{5} C \][/tex]
3. Next, multiply both sides by [tex]\( \frac{5}{9} \)[/tex] to solve for [tex]\( C \)[/tex]:
[tex]\[ C = \frac{5}{9} (F - 32) \][/tex]
This inverse equation allows us to convert Fahrenheit back into Celsius.
The equation [tex]\( C = \frac{5}{9} (F - 32) \)[/tex] is the inverse of the original function, and yes, it is a function because, for each input [tex]\( F \)[/tex], there is exactly one output [tex]\( C \)[/tex].
b. Using the inverse to find the Celsius temperature for [tex]\( 16^\circ F \)[/tex]:
Now, let's use the inverse formula to find the Celsius temperature that corresponds to [tex]\( 16^\circ F \)[/tex].
1. Use the inverse formula:
[tex]\[ C = \frac{5}{9} (F - 32) \][/tex]
2. Substitute [tex]\( F = 16 \)[/tex] into the formula:
[tex]\[ C = \frac{5}{9} (16 - 32) \][/tex]
3. Simplify inside the parentheses:
[tex]\[ 16 - 32 = -16 \][/tex]
4. Then calculate:
[tex]\[ C = \frac{5}{9} \times (-16) \][/tex]
5. Compute the product:
[tex]\[ C = -8.88888888888889 \][/tex]
Therefore, the Celsius temperature that corresponds to [tex]\( 16^\circ F \)[/tex] is approximately [tex]\(-8.89^\circ C\)[/tex].
a. Finding the inverse of the formula:
The original formula for converting from Celsius to Fahrenheit is:
[tex]\[ F = \frac{9}{5} C + 32 \][/tex]
To find the inverse of this formula, we want to solve for [tex]\( C \)[/tex] in terms of [tex]\( F \)[/tex]. This involves isolating [tex]\( C \)[/tex] on one side of the equation.
1. Start with the equation:
[tex]\[ F = \frac{9}{5} C + 32 \][/tex]
2. To isolate [tex]\( C \)[/tex], first subtract 32 from both sides:
[tex]\[ F - 32 = \frac{9}{5} C \][/tex]
3. Next, multiply both sides by [tex]\( \frac{5}{9} \)[/tex] to solve for [tex]\( C \)[/tex]:
[tex]\[ C = \frac{5}{9} (F - 32) \][/tex]
This inverse equation allows us to convert Fahrenheit back into Celsius.
The equation [tex]\( C = \frac{5}{9} (F - 32) \)[/tex] is the inverse of the original function, and yes, it is a function because, for each input [tex]\( F \)[/tex], there is exactly one output [tex]\( C \)[/tex].
b. Using the inverse to find the Celsius temperature for [tex]\( 16^\circ F \)[/tex]:
Now, let's use the inverse formula to find the Celsius temperature that corresponds to [tex]\( 16^\circ F \)[/tex].
1. Use the inverse formula:
[tex]\[ C = \frac{5}{9} (F - 32) \][/tex]
2. Substitute [tex]\( F = 16 \)[/tex] into the formula:
[tex]\[ C = \frac{5}{9} (16 - 32) \][/tex]
3. Simplify inside the parentheses:
[tex]\[ 16 - 32 = -16 \][/tex]
4. Then calculate:
[tex]\[ C = \frac{5}{9} \times (-16) \][/tex]
5. Compute the product:
[tex]\[ C = -8.88888888888889 \][/tex]
Therefore, the Celsius temperature that corresponds to [tex]\( 16^\circ F \)[/tex] is approximately [tex]\(-8.89^\circ C\)[/tex].
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