3) A website heis 300,000 members. the number of Members (y) increases by $$ 11 \% $$ each year. select all that apply d(A. this situation can Be modeled by the equation $$ 1=300,000(1.11)^{+} $$ B. B. 0 this situation can be modeled by the equation $$ 300,000(0,11)^{+} $$ Y=300,000 (0.11) Y. It this sitation cain be modered by the equation $$ Y=0.11(300,000)^{t} $$ D. Ithere will Be approximately 33,000 members after 4 years Be a E. a thene w after 4 years F. I there will be approximately 455000 members after 4 years

Question

UpStudy AI · Accepted Answer

Let's analyze each of the provided options based on the information that the website starts with **300,000 members** and the number of members increases by **11% each year**.

### 1. **Modeling the Situation**

To model the growth of members, we use the **exponential growth formula**:

$$
Y(t) = Y_0 \times (1 + r)^t
$$

Where:
- $$ Y(t) $$ is the number of members after $$ t $$ years.
- $$ Y_0 $$ is the initial number of members.
- $$ r $$ is the growth rate (expressed as a decimal).
- $$ t $$ is the time in years.

Given:
- $$ Y_0 = 300,000 $$
- $$ r = 11\% = 0.11 $$

So, the correct model is:

$$
Y(t) = 300,000 \times (1.11)^t
$$

### 2. **Evaluating the Options**

Let's evaluate each option based on the correct model.

#### **Option A**
> **A.** This situation can be modeled by the equation $$ Y = 300,000(1.11)^t $$.

**Evaluation:** *Correct.* This matches the exponential growth model correctly.

#### **Option B**
> **B.** This situation can be modeled by the equation $$ Y = 300,000(0.11)^t $$.  
> $$ Y = 300,000 \times 0.11 $$

**Evaluation:** *Incorrect.* Using $$ 0.11 $$ as the base of the exponent implies a decay model, which is not appropriate here.

#### **Option C**
> **C.** This situation can be modeled by the equation $$ Y = 0.11(300,000)^t $$.

**Evaluation:** *Incorrect.* This incorrectly applies the growth rate and misplaces the initial value in the exponent.

#### **Option D**
> **D.** There will be approximately 33,000 members after 4 years.

**Evaluation:** *Incorrect.* Starting with 300,000 members and increasing by 11% each year, the number of members after 4 years is much higher.

#### **Option F**
> **F.** There will be approximately 455,000 members after 4 years.

**Evaluation:** *Correct.*

**Calculation:**
$$
Y(4) = 300,000 \times (1.11)^4 \approx 300,000 \times 1.5171 \approx 455,130
$$

### **Summary of Correct Options**

- **A.** Correct
- **F.** Correct

All other options are either incorrect or incomplete.

**Final Answer:**  
**Options A and F are correct.**
**Correct Options:** A. This situation can be modeled by the equation $$ Y = 300,000(1.11)^t $$. F. There will be approximately 455,000 members after 4 years.

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Solución

1. Modeling the Situation

2. Evaluating the Options

Option A

Option B

Option C

Option D

Option F

Summary of Correct Options

Bonus Knowledge

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