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

We want to combine b(t) and t(h) to create a function that described the number of bacteria, b, as a function of time, h. 
This function can be called b(t(h)), or b(h).
Consider the function b(t), or bacteria as a function of temperature. 
b(t) = 20t² - 70t +300
This describes the number of bacteria, given the temperature.

t(h) described the temperature as a function of time, specifically hours after refrigeration:
t(h) = 2h + 3

Since the time h can tell us the temperature t, and the temperature t can tell us the # of bacteria b, we can create a function that tells us the number of bacteria, b, given hours following refrigeration, h.

To do this, we plug t(h) in for every t in the b(t) function:
b(t(h)) = 20 (2h+3)² - 70(2h+3) + 300
We can also call this function b(h), since we now can express b as a function of h.

Simplify the function:
b(h) = 20 (4h²+12h+9)-14h-210+300
b(h) = 80h² +100h +270

The Answer is B

Answer 2

Given:
b(t) = 20t² - 70t + 300
t(h) = 2h + 3

Therefore
b(h) = 20(2h + 3)² - 70(2h + 3) + 300
       = 20(4h² + 12h + 9) - 140h - 210 + 300
       = 80h² + 240h + 180 - 140h - 210 + 300
       = 80h² + 100h + 270

Answer:  b(h) = 80h² + 100h + 270