Exponential Growth And Decay Word Problems Worksheet With Answers ... - Free Printable

You've uploaded a worksheet titled "Exponential Growth and Decay Word Problems" with two problems solved by hand. Let's go through each problem, verify the solutions, and explain them clearly.

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Problem 1: Florida’s Population Growth

Given:
- In 2000, Florida’s population was 16 million.
- The population grows at 2% per year (exponentially).
- Find the population in 2006.

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#### Step-by-step Solution:

The general formula for exponential growth is:

$$
y = a(1 + r)^x
$$

Where:
- $ y $ = final amount
- $ a $ = initial amount (starting value)
- $ r $ = growth rate (as a decimal)
- $ x $ = time in years

Plug in the values:
- $ a = 16,000,000 $
- $ r = 2\% = 0.02 $
- $ x = 2006 - 2000 = 6 $ years

$$
y = 16,000,000 \times (1 + 0.02)^6
$$
$$
y = 16,000,000 \times (1.02)^6
$$

Now calculate $ (1.02)^6 $:

$$
(1.02)^6 \approx 1.126162
$$

$$
y \approx 16,000,000 \times 1.126162 = 18,018,598.71
$$

Rounded to the nearest person:
$$
\boxed{18,018,599} \text{ people}
$$

✔ Your answer of 18,018,598.71 ≈ 18,018,599 is correct.

> ✔️ Answer: Florida’s population in 2006 was approximately 18,018,599 people.

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Problem 2: Hospital Costs in the U.S.

Given:
- In 1995, average daily hospital cost = $968
- Cost increases by 4% per year (exponentially)
- Model this growth and estimate the cost in 2010

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#### Part a) Write an equation to model the cost since 1995

Using the same exponential growth formula:

$$
y = a(1 + r)^x
$$

- $ a = 968 $
- $ r = 4\% = 0.04 $
- $ x $ = number of years since 1995

So the equation is:

$$
\boxed{y = 968(1.04)^x}
$$

✔ Your answer is correct.

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#### Part b) Estimate the cost per day in 2010

First, find $ x $:

$$
x = 2010 - 1995 = 15 \text{ years}
$$

Now plug into the equation:

$$
y = 968 \times (1.04)^{15}
$$

Calculate $ (1.04)^{15} $:

$$
(1.04)^{15} \approx 1.800943
$$

Now multiply:

$$
y \approx 968 \times 1.800943 \approx 1,743.3133
$$

Round to the nearest hundredth (i.e., two decimal places):

$$
\boxed{1,743.31}
$$

✔ Your calculation is correct.

> ✔️ Answer: The estimated cost per day in 2010 is $1,743.31

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✔ Final Answers Summary:

1. Florida’s population in 2006:
$$
\boxed{18,018,599} \text{ people}
$$

2. Hospital cost model:
$$
\boxed{y = 968(1.04)^x}
$$

3. Estimated hospital cost in 2010:
$$
\boxed{\$1,743.31} \text{ per day}
$$

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🔍 Explanation Recap:

- Exponential growth uses $ y = a(1 + r)^x $
- Always identify:
- Starting amount ($ a $)
- Growth rate ($ r $ as decimal)
- Time period ($ x $)
- Use calculator for powers like $ (1.02)^6 $ or $ (1.04)^{15} $
- Round appropriately based on instructions (e.g., nearest hundredth)

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✔ All your work is accurate and well-explained! Great job solving both problems correctly.