Question

The equation $C = \dfrac{5}{9}(F - 32)$ shows how temperature $F$, measured in degrees Fahrenheit, relates to temperature $C$ measured in degrees Celsius. Based on this equation, which of the given statements is correct?
(I) A temperature increase of ${1^o}F$ is equivalent to a temperature increase of ${\dfrac{5}{9}^o}C$
(II) A temperature increase of ${1^o}C$ is equivalent to temperature increase of ${1.8^o}F$
(III) A temperature increase of ${\dfrac{5}{9}^o}F$ is equivalent to temperature increase of ${1^o}C$
A. I only
B. II only
C. III only
D. I and II only

Answer 1

In order to solve this question, we will use the given equation having relation of $F$ and $C$ scales of temperature. We will put the values of $C$ and $F$ according to the given statements in each of three cases and then figure out which option is the correct one.

Complete step by step answer:
Here, C and F will represent initial values of Celsius and Fahrenheit scales while C’ and F; be the modified values of F and C according to given information in each of three statements.From statement (I) we have, $F' = 1 + F$ one degree rise in Fahrenheit scale, let us assume new Celsius scale be C’ and initial values C and F are related as given equation $C = \dfrac{5}{9}(F - 32)$ so, new values are related as
$C' = \dfrac{5}{9}(F' - 32)$
On putting value of $F' = 1 + F$ we get,
$\Rightarrow C' = \dfrac{5}{9}(F + 1 - 32)$
$\Rightarrow C' = \dfrac{5}{9}(F - 32) + \dfrac{5}{9}$
We can write it as,
$C' = C + \dfrac{5}{9}$
Hence, statement (I) A temperature increase of ${1^o}F$ is equivalent to a temperature increase of ${\dfrac{5}{9}^o}C$ is correct one.

From statement (II) we have, $C' = C + 1$ one degree rise in Celsius scale, so again using relation for new values of F and c we have, also we can rearrange the given equation $C = \dfrac{5}{9}(F - 32)$ in the form of $F = \dfrac{9}{5}C + 32$ so,
$\Rightarrow F' = \dfrac{9}{5}C' + 32$
And putting the value of $C' = C + 1$ we get,
$\Rightarrow F' = \dfrac{9}{5}(C + 1) + 32$
$\Rightarrow F' = [\dfrac{9}{5}C + 32] + \dfrac{9}{5}$
or we can write it as,
$F' = F + 1.8$
Hence, statement (II) A temperature increase of ${1^o}C$ is equivalent to a temperature increase of ${1.8^o}F$ is the correct one.

Now, for statement (III) we have, $F' = \dfrac{5}{9} + F$ so putting this value in equation $C' = \dfrac{5}{9}(F' - 32)$ we get,
$\Rightarrow C' = \dfrac{5}{9}(\dfrac{5}{9} + F - 32)$
$\Rightarrow C' = \dfrac{5}{9}(F - 32) + \dfrac{5}{9} \times \dfrac{5}{9}$
We can write it as,
$C' = C + \dfrac{{25}}{{81}}$
Hence , the statement (III) A temperature increase of ${\dfrac{5}{9}^o}F$ is equivalent to a temperature increase of ${1^o}C$ is incorrect one.

Hence, the correct option is D.

Note: It should be remembered that, Celsius and Fahrenheit are the two most common units in which temperature is measured and the given equation is generally used to convert temperature from one unit to another between Celsius and Fahrenheit scales.