Question

evaluate
name:
□ complete the following problems. show all your work
□ complete the lesson reflections for each section by circling your current understanding of the learning goal(s).
□ complete the following problems. show all your work
learning goal from lesson 1 & 4\tlesson reflection (circle one)
☑ i can construct exponential functions given a graph, a description of a relationship, or two input - output pairs.\ti got it!\tim still learning it

1. write an exponential function that includes the following points (2,32) and (3,64). (lesson 1) (1 point)
2. martha is making fresh yogurt with a new sample containing 200,000 healthy, probiotic bacteria at the start (at t = 0). after putting the sample into her yogurt maker, she records the total number of probiotic bacteria f(t) in the sample as it grows exponentially each hour (t).

choose two input - output pairs from the data table to help you determine the function that models this real - world situation. (1 point)
t\tf(t)
0\t200,000
1\t276,000
2\t380,880
3\t525,614

1. how many probiotic bacteria will be in the sample after 24 hours? (lesson 1) (1 point)

Explanation:

Step1: Recall the general form of an exponential function

The general form of an exponential function is $y = ab^{x}$, where $a$ is the initial - value and $b$ is the base.

Step2: Solve for the exponential function in problem 1

Given the points $(2,32)$ and $(3,64)$:
Substitute into $y = ab^{x}$:
For the point $(2,32)$: $32=ab^{2}$ (Equation 1)
For the point $(3,64)$: $64 = ab^{3}$ (Equation 2)
Divide Equation 2 by Equation 1: $\frac{ab^{3}}{ab^{2}}=\frac{64}{32}$, which simplifies to $b = 2$.
Substitute $b = 2$ into Equation 1: $32=a\times2^{2}$, so $32 = 4a$, and $a = 8$.
The exponential function is $y = 8\times2^{x}$.

Step3: Solve for the exponential function in problem 2

The general form of the exponential function is $f(t)=ab^{t}$.
Using the input - output pairs $(0,200000)$ and $(1,276000)$:
When $t = 0$, $f(0)=ab^{0}=a$, so $a = 200000$.
When $t = 1$, $f(1)=ab^{1}=200000b$. Since $f(1)=276000$, then $200000b=276000$, and $b=\frac{276000}{200000}=1.38$.
The function is $f(t)=200000\times1.38^{t}$.

Step4: Solve for the number of bacteria in problem 3

We have the function $f(t)=200000\times1.38^{t}$.
When $t = 24$, $f(24)=200000\times1.38^{24}$.
Using a calculator, $1.38^{24}\approx10777.77$, and $f(24)=200000\times10777.77 = 2155554000$.

Answer:

1. $y = 8\times2^{x}$
2. Using $(0,200000)$ and $(1,276000)$, the function is $f(t)=200000\times1.38^{t}$
3. $2155554000$