Math / Algebra

Question7 Mark for Review
A model predicts that the population of Springfield was 15,000 in 2005. The model also predicts that each year for the next 5 years, the population $p$ increased by $4 \%$ of the previous year's population. Which equation best represents this model, where $x$ is the number of years after 2005 , for $x \leq 5$ ? (A) $p=0.96(15,000)^{x}$ (B) $p=1.04(15,000)^{x}$ (C) $p=15,000(0.96)^{x}$ (D) $p=15,000(1.04)^{x}$ Question 7 of 22

Studdy Solution

STEP 1

1. The initial population of Springfield in 2005 is 15,000.
2. The population increases by 4% each year for the next 5 years.
3. We need to find an equation that models this population growth.
4. The variable $x$ represents the number of years after 2005, where $x \leq 5$ .

STEP 2

1. Understand the concept of exponential growth.
2. Determine the growth factor.
3. Write the exponential growth equation.
4. Compare with given options.

STEP 3

Understand the concept of exponential growth.
In exponential growth, the population increases by a constant percentage each year. This means the population for each subsequent year is a fixed multiple of the previous year's population.

STEP 4

Determine the growth factor.
Since the population increases by 4% each year, the growth factor is $1 + 0.04 = 1.04$ .

STEP 5

Write the exponential growth equation.
The general form of an exponential growth equation is $p = p_0 \cdot (1 + r)^x$ , where $p_0$ is the initial population, $r$ is the growth rate, and $x$ is the number of years.
For this problem, $p_0 = 15,000$ , $r = 0.04$ , and the growth factor is $1.04$ .
Thus, the equation is: $p = 15,000 \cdot (1.04)^x$

STEP 6

Compare with given options.
The equation $p = 15,000 \cdot (1.04)^x$ matches option (D).
The correct equation is: $\boxed{p = 15,000 \cdot (1.04)^x}$

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Studdy Solution

STEP 1

1. The initial population of Springfield in 2005 is 15,000.
2. The population increases by 4% each year for the next 5 years.
3. We need to find an equation that models this population growth.
4. The variable $x$ represents the number of years after 2005, where $x \leq 5$ .

STEP 2

1. Understand the concept of exponential growth.
2. Determine the growth factor.
3. Write the exponential growth equation.
4. Compare with given options.

STEP 3

Understand the concept of exponential growth.
In exponential growth, the population increases by a constant percentage each year. This means the population for each subsequent year is a fixed multiple of the previous year's population.

STEP 4

Determine the growth factor.
Since the population increases by 4% each year, the growth factor is $1 + 0.04 = 1.04$ .

STEP 5

STEP 6

Compare with given options.
The equation $p = 15,000 \cdot (1.04)^x$ matches option (D).
The correct equation is: $\boxed{p = 15,000 \cdot (1.04)^x}$

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Studdy Solution

STEP 1

1. The initial population of Springfield in 2005 is 15,000.2. The population increases by 4% each year for the next 5 years.3. We need to find an equation that models this population growth.4. The variable x x x represents the number of years after 2005, where x≤5 x \leq 5 x≤5.

STEP 2

1. Understand the concept of exponential growth.2. Determine the growth factor.3. Write the exponential growth equation.4. Compare with given options.

STEP 3

Understand the concept of exponential growth.In exponential growth, the population increases by a constant percentage each year. This means the population for each subsequent year is a fixed multiple of the previous year's population.

STEP 4

Determine the growth factor.Since the population increases by 4% each year, the growth factor is 1+0.04=1.04 1 + 0.04 = 1.04 1+0.04=1.04.

STEP 5

STEP 6

Compare with given options.The equation p=15,000⋅(1.04)x p = 15,000 \cdot (1.04)^x p=15,000⋅(1.04)x matches option (D). The correct equation is: p=15,000⋅(1.04)x \boxed{p = 15,000 \cdot (1.04)^x} p=15,000⋅(1.04)x​

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Studdy solves anything!

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. The initial population of Springfield in 2005 is 15,000.
2. The population increases by 4% each year for the next 5 years.
3. We need to find an equation that models this population growth.
4. The variable $x$ represents the number of years after 2005, where $x \leq 5$ .

1. Understand the concept of exponential growth.
2. Determine the growth factor.
3. Write the exponential growth equation.
4. Compare with given options.

Understand the concept of exponential growth.
In exponential growth, the population increases by a constant percentage each year. This means the population for each subsequent year is a fixed multiple of the previous year's population.

Determine the growth factor.
Since the population increases by 4% each year, the growth factor is $1 + 0.04 = 1.04$ .

Compare with given options.
The equation $p = 15,000 \cdot (1.04)^x$ matches option (D).
The correct equation is: $\boxed{p = 15,000 \cdot (1.04)^x}$