Suppose a tap mixes hot water and cold water in a ratio that depends linearly on the proportion of opening. Water out of the tap has temperature 40°C when the tap is half-open, and 30°C when it is three-fourths open. To get water at 50°C, the tap should be _____.

Correct option is B

Solution: Since temperature varies linearly with the proportion of opening, let:
Temperature = a × (opening) + b
Let opening be measured as a fraction.
Given:
· At opening = 1/2, temperature = 40
· At opening = 3/4, temperature = 30
Form two equations:
40 = a(1/2) + b 30 = a(3/4) + b
Subtracting the second from the first:
10 = a(1/2 − 3/4) 10 = a(−1/4)
So, a = −40
Substitute in first equation:
40 = (−40)(1/2) + b 40 = −20 + b b = 60
So, Temperature = −40(opening) + 60
Now, for temperature = 50:
50 = −40(opening) + 60 40(opening) = 10 opening = 1/4
Correct answer: (b) one-fourth open