The problem states that the number of mosquitoes per cubic meter is a linear function of the temperature. If we take #M# to represent mosquitoes (in units of #1/m^3#) and #T# to represent temperature (in units of #C#), then we can write a function of #M# as a linear equation:
#M(T) = kT + D#
#k# represents some constant multiplier we do not yet know, and #D# represents some possible constant adjustment to the calculated value of #kT#. #D# may be 0. (I'm using #D# here instead of a more common #C# for an unknown constant to avoid confusion with the celsius units.)
We are given two initial conditions in the problem in the first two sentences, which converted carefully to our function can give us enough information to determine both #k# and #D#:
#{(M(14) = 3.6,=>,3.6 = k(14) + D,[1]),(M(40) = 7.5,=>,7.5=k(40) + D,[2]):} #
We can now solve the equations on the right simultaneously. I'll opt to subtract equation [1] from equation [2]:
#7.5 - 3.6 = k(40) - k(14) #
#3.9 = 26k #
#k = 0.15#
Substituting this value of #k# into either equation will yield #D#. I'll use [1]:
#3.6 = k(14) + D #
#3.6 = (0.15)(14) + D #
#3.6 = 2.1 + D#
# D = 1.5#
The final function is #M(T) = 0.15T + 1.5#
To predict the mosquito population at temperature #T = 22#, we can substitute #T=22# into this function:
#M(22) = 0.15(22) + 1.5 = 3.3 + 1.5 = 4.8#
