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Question: The speed of the sound in an air wave depends on temperature and is given by the relation \(a=1052+1...

The speed of the sound in an air wave depends on temperature and is given by the relation a=1052+1.08ta=1052+1.08t, where a is the speed of the sound in feet per second and t is the temperature in degree Fahrenheit. At which temperature (in degrees Fahrenheit) will the speed of the sound be closest to 1000 feet per second
[a] -46
[b] -48
[c] -49
[d] -50

Explanation

Solution

Find the value of the speed of the sound at −46∘F,−48∘F,−49∘F-46{}^\circ F,-48{}^\circ F,-49{}^\circ F and −50∘F-50{}^\circ Fusing the expression for speed of sound in terms of t given in the question. Determine which value is closest to 1,000 feet per second. Hence determine the temperature at which the speed of sound is closest to 1,000 feet per second.

Complete step-by-step answer :
We shall solve the question by finding the speed of the sound at −46∘F,−48∘F,−49∘F-46{}^\circ F,-48{}^\circ F,-49{}^\circ F and −50∘F-50{}^\circ Fusing the expression for speed of sound in terms of t given in the question.
At t=−46∘Ft=-46{}^\circ F, we have
a=1052+1.08(−46)=1052−49.68=1002.32a=1052+1.08\left( -46 \right)=1052-49.68=1002.32
Hence the speed of the sound is 1002.32 feet per second, which is 1002.32−1000=2.321002.32-1000=2.32 feet per second more than 1000 feet per second
At t=−48∘Ft=-48{}^\circ F, we have
a=1052+1.08(−48)=1052−51.84=1000.16a=1052+1.08\left( -48 \right)=1052-51.84=1000.16
Hence the speed of the sound is 1000.16 feet per second, which is 1000.16−1000=0.161000.16-1000=0.16 feet per second more than 1000 feet per second
At t=−49∘Ft=-49{}^\circ F, we have
a=1052+1.08(−49)=1052−52.92=999.08a=1052+1.08\left( -49 \right)=1052-52.92=999.08
Hence the speed of the sound is 999.08 feet per second, which is 1000−999.08=0.921000-999.08=0.92 feet per second less than 1000 feet per second
At t=−50∘Ft=-50{}^\circ F, we have
a=1052+1.08(−50)=1052−54=998a=1052+1.08\left( -50 \right)=1052-54=998
Hence the speed of the sound is 998 feet per second, which is 1000−998=21000-998=2 feet per second less than 1000 feet per second.
Hence the speed of the sound is closest to 1000 feet per second when temperature is −48∘F-48{}^\circ F
Hence option [b] is correct.

Note : [1] In these types of questions students make mistake by considering only the values greater than the target value to be closer to that which is incorrect since the value whose absolute value of difference with target value is smallest is the closest one and can be larger or smaller or equal to the target value.