EXPONENTIAL EQUATIONS & FUNCTIONS

Question

EXPONENTIAL EQUATIONS & FUNCTIONS
1)A culture of bacteria doubles in weight approximately every 2.5 hours. The number of bacteria after t hours later can be approximated by 𝑁(𝑡) = 175(2.5)𝑡
a) How many bacteria would there be initially (𝑡 = 0)?

2)Solve for x.
a) 43−2𝑥 = 64 b) 25𝑥+1 = 53𝑥
b) Find 𝑁(4). What does this number tell you in terms of this problem?

 

 

Answer 1

Courtney:

These growth problem always involve an exponent, i.e., a power of time, t.

The comment is correct, except that we also need a growth factor, k.

So, N = 175 x 2^kt/2,5 . . . . . . . (1)

Since it doubles every 2.5 hours, we can write: 2N = 175 x 2.5^k(t+2.5)/2.5 . . . . (2)

Divide (1) by (2): 2 = 2^((kt / 2.5) + k) - kt/2.5) . . . . subtract exponents

= 2^k

=> k =1

The rest I leave for you as an exercise.

So, our growth equation is: N(t) = 175 x 2^t/2.5

Initially, t = 0 => N(0) = 175 x 2⁰

= 175