Question

earthquake problem
write an exponential function to model earthquake intensity as a function of a richter scale number. how can you use your function to compare the intensity of the 1811 new madrid and 1906 san francisco earthquakes?

• san francisco earthquake (1906): 7.8 on richter scale
• new madrid earthquake (1811): 8.1 on richter scale

richter scale	earthquake intensity
2	100
3	1,000
4	10,000

algae problem
the table shows the number of algae cells in pool water samples. a pool will turn green when there are 24 million alae cells or more. write and graph an exponential function to model the expected number of algae cells as a function of the number of days. if the pattern continues, in how many days will the water turn green?

day	number of algae cells
1	10,000
2	50,000
3	250,000
4	1,250,000

Explanation:

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Earthquake Problem

Step1: Define exponential function

From the table, intensity $I$ follows $I(R) = 10^R$, where $R$ = Richter number.

Step2: Calculate SF earthquake intensity

Substitute $R=7.8$:
$I(7.8) = 10^{7.8}$

Step3: Calculate NM earthquake intensity

Substitute $R=8.1$:
$I(8.1) = 10^{8.1}$

Step4: Find intensity ratio

Divide NM intensity by SF intensity:
$\frac{I(8.1)}{I(7.8)} = 10^{8.1-7.8} = 10^{0.3} \approx 2$

Step1: Identify exponential form

The general form is $A(d) = A_0 \cdot b^d$, where $A_0=2000$ (initial cells), $d$ = days.

Step2: Find growth factor $b$

Use day 1 data: $10000 = 2000b^1 \implies b = \frac{10000}{2000}=5$
Model: $A(d)=2000 \cdot 5^d$

Step3: Set up equation for 24M cells

$2000 \cdot 5^d = 24000000$

Step4: Simplify and solve for $d$

Divide both sides by 2000:
$5^d = 12000$
Take log base 5:
$d = \log_5(12000) = \frac{\ln(12000)}{\ln(5)} \approx 6.1$

Answer:

The exponential model is $I(R)=10^R$. The 1811 New Madrid earthquake (Richter 8.1) was approximately 2 times as intense as the 1906 San Francisco earthquake (Richter 7.8).

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Algae Problem