Question

The temperature in degree Fahrenheit $\left({}^{\circ }F\right)$ ranges between$86\mathrm{and}95$ on a certain island. Celsius $\left({}^{\circ }C\right)$ temperature is related to Fahrenheit temperature by the formula $C=\frac{5}{9}\left(F-32\right)$. What is the Celsius temperature range of this island.

Answer 1

Step 1:

Find Fahrenheit temperature using the given formula:

The Fahrenheit range of temperature is,

${86}^{\circ }<F<{95}^{\circ }$

Determine the formula for Fahrenheit temperature.

$\begin{array}{rcl}C& =& \frac{5}{9}\left(F-32\right)\\ & \Rightarrow & 9C=5\left(F-32\right)\\ & \Rightarrow & 9C=5F-160\\ & \Rightarrow & 9C+160=5F\\ & \Rightarrow & F=\frac{9C+160}{5}\\ \therefore F& =& \frac{9C}{5}+32\end{array}$

Step 2:

Find the Celsius temperature range of the island:
$\begin{array}{rcl}86& <& F<95\\ & \Rightarrow & 86<\frac{9C}{5}+32<95\\ & \Rightarrow & \left(86\times 5\right)<9C+\left(32\times 5\right)<\left(95\times 5\right)\\ & \Rightarrow & 430<9C+160<475\\ & \Rightarrow & 430-160<9C<475-160\\ & \Rightarrow & 270<9C<315\\ & \Rightarrow & \frac{270}{9}<C<\frac{315}{9}\\ \therefore {30}^{\circ }& <& C<{35}^{\circ }\end{array}$
Thus, the required range of the temperature is $\begin{array}{rcl}{30}^{\circ }& <& C<{35}^{\circ }\end{array}$.