In an experiment, a solution of hydrochloric acid is to be kept between 30∘ and 35∘ Celsius. What is the range of temperature in degree Fahrenheit if conversion formula is given by C=95( F−32), where C and F represent temperature in degree Celsius and degree Fahrenheit, respectively.
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Step-by-Step Solution
Step 1: Set up the inequality for Celsius temperature
The problem states that the temperature in Celsius, denoted by C, must be kept between 30∘ and 35∘. This can be written as a compound inequality.
Step 2: Substitute the conversion formula
We are given the conversion formula C=95(F−32). We substitute this expression for C into the inequality to relate the Fahrenheit temperature F to the given Celsius range.
Step 3: Multiply by the reciprocal to isolate (F-32)
To isolate the term (F−32), we multiply all parts of the inequality by the reciprocal of 95, which is 59. This operation maintains the direction of the inequality signs because we are multiplying by a positive number.
Step 4: Simplify the inequality
Now, we perform the multiplication on both sides of the inequality. 30×59=6×9=54 and 35×59=7×9=63.
Step 5: Add 32 to all parts to find the range of F
To find the range of F, we add 32 to all parts of the inequality. This isolates F and gives us the temperature range in Fahrenheit.
Step 6: Calculate the final Fahrenheit range
Performing the addition, we get 54+32=86 and 63+32=95. Thus, the temperature in Fahrenheit must be between 86∘ and 95∘.