Question

The number of bacteria (b) in a batch of refrigerated food is given by the function b(t) = 20t2 − 70t + 300, where t is the temperature of the food in degrees Fahrenheit. The function t(h) = 2h + 3 gives the temperature of the food h hours after it is removed from refrigeration. Which expression correctly represents the number of bacteria (b) in the food as a function of the number of hours (h) the food is unrefrigerated?

Answer 1

$b(t)=20t^2-70t+300$ represent the number of bacteria in a batch of refrigerated food, and $t(h)$ represent the temperature of the food h hours after it is removed from refrigeration. To find the number of bacteria as function of the number of hours the food is refrigerated, we are going to find the composite function (b ∘ t). To do it, we are going to evaluate our first functions $b(t)$ at our second function $t(h)$:
$b(t)=20t^2-70t+300$
$t(h)=2h+3$
$b(2h+3)=20(2h+3)^2-70(2h+3)+300$
$b(2h+3)=20(4h^2+12h+9)-140h-210+300$
$b(2h+3)=80h^2+240h+180-140h+90$
$b(2h+3)=80h^2+100h+270$

We can conclude that the expression that represent the number of bacteria (b) in the food as a function of the number of hours (h) the food is unrefrigerated is $b(h)=80h^2+100h+270$