Examkin

Soru

Zorluk: OrtaExponential Functions and Equations

The table below shows the population of a colony of bacteria, P(t)P(t), at various times tt, in hours.

Time tt (hours)Population P(t)P(t)
0120
3960
67,680

If the population can be modeled by an exponential function of the form P(t)=a⋅btP(t) = a \cdot b^t, where aa and bb are constants, what is the value of bb?

  1. 2Cevap
  2. B
    8
  3. C
    4
  4. D
    83\frac{8}{3}

Cevap

2
The correct answer is the value that represents the hourly growth factor of the bacteria. Since the initial population at t=0t = 0 is 120, we have a=120a = 120. Using the population at t=3t = 3, we set up the equation 120⋅b3=960120 \cdot b^3 = 960. Dividing both sides by 120 yields b3=8b^3 = 8. Taking the cube root of 8 gives b=2b = 2. We can verify this with the third data point: 120⋅26=120⋅64=7,680120 \cdot 2^6 = 120 \cdot 64 = 7,680, which matches the table.

Adım Adım Çözüm

1
Find the value of the constant aa using the initial value from the table.
a=120a = 120
When t=0t = 0, the population P(0)=120P(0) = 120. Substituting these values into P(t)=a⋅btP(t) = a \cdot b^t gives 120=a⋅b0120 = a \cdot b^0, and since b0=1b^0 = 1, we get a=120a = 120.
2
Set up an equation to solve for bb using another data point from the table.
120⋅b3=960120 \cdot b^3 = 960
When t=3t = 3, the population P(3)=960P(3) = 960. Substituting a=120a = 120, t=3t = 3, and P(3)=960P(3) = 960 into the function gives 960=120⋅b3960 = 120 \cdot b^3.
3
Isolate b3b^3 and solve for bb by taking the cube root.
b=2b = 2
Dividing both sides of 120⋅b3=960120 \cdot b^3 = 960 by 120120 gives b3=8b^3 = 8. Taking the cube root of both sides gives b=83=2b = \sqrt[3]{8} = 2.

Anahtar Kavram

Determining parameters of exponential functions from a table of values.
Bu soruyu puanla

Konular

Exponential Functions and Equations

Bu konudan daha fazla soru