To solve this problem, we need to set up an equation that models the exponential growth of the elephant's weight and then solve for the number of months, $ t $, it will take for the elephant to weigh 675 pounds.

### Step-by-Step Solution:

1. **Understand the Problem:**
- Initial weight of the elephant: 200 pounds.
- Each month, the weight increases by half of the previous month's weight.
- We need to find the number of months, $ t $, for the elephant to weigh 675 pounds.

2. **Set Up the Exponential Growth Model:**
- Let $ W(t) $ be the weight of the elephant after $ t $ months.
- The weight increases by half of the previous month's weight, which means the weight is multiplied by $ 1.5 $ each month.
- The initial weight is 200 pounds.

Therefore, the weight after $ t $ months can be modeled by the equation:
$$
W(t) = 200 \times (1.5)^t
$$

3. **Set Up the Equation to Solve for $ t $:**
- We want to find $ t $ when the weight $ W(t) $ is 675 pounds.
$$
200 \times (1.5)^t = 675
$$

4. **Solve for $ t $:**
- Divide both sides of the equation by 200 to isolate the exponential term:
$$
(1.5)^t = \frac{675}{200}
$$
$$
(1.5)^t = 3.375
$$

- Take the natural logarithm (ln) of both sides to solve for $ t $:
$$
\ln((1.5)^t) = \ln(3.375)
$$

- Use the property of logarithms that $ \ln(a^b) = b \ln(a) $:
$$
t \ln(1.5) = \ln(3.375)
$$

- Solve for $ t $:
$$
t = \frac{\ln(3.375)}{\ln(1.5)}
$$

5. **Calculate the Value of $ t $:**
- Use a calculator to find the natural logarithms:
$$
\ln(3.375) \approx 1.216
$$
$$
\ln(1.5) \approx 0.405
$$

- Divide the logarithms:
$$
t = \frac{1.216}{0.405} \approx 3
$$

### Conclusion:
- The equation that can be solved to determine the number of months $ t $ is:
$$
200 \times (1.5)^t = 675
$$

- It will take approximately 3 months for the elephant to weigh 675 pounds.

Question

To solve this problem, we need to set up an equation that models the exponential growth of the elephant's weight and then solve for the number of months, $ t $, it will take for the elephant to weigh 675 pounds.

### Step-by-Step Solution:

1. **Understand the Problem:**
   - Initial weight of the elephant: 200 pounds.
   - Each month, the weight increases by half of the previous month's weight.
   - We need to find the number of months, $ t $, for the elephant to weigh 675 pounds.

2. **Set Up the Exponential Growth Model:**
   - Let $ W(t) $ be the weight of the elephant after $ t $ months.
   - The weight increases by half of the previous month's weight, which means the weight is multiplied by $ 1.5 $ each month.
   - The initial weight is 200 pounds.

Therefore, the weight after $ t $ months can be modeled by the equation:
   $$
   W(t) = 200 \times (1.5)^t
   $$

3. **Set Up the Equation to Solve for $ t $:**
   - We want to find $ t $ when the weight $ W(t) $ is 675 pounds.
   $$
   200 \times (1.5)^t = 675
   $$

4. **Solve for $ t $:**
   - Divide both sides of the equation by 200 to isolate the exponential term:
   $$
   (1.5)^t = \frac{675}{200}
   $$
   $$
   (1.5)^t = 3.375
   $$

- Take the natural logarithm (ln) of both sides to solve for $ t $:
   $$
   \ln((1.5)^t) = \ln(3.375)
   $$

- Use the property of logarithms that $ \ln(a^b) = b \ln(a) $:
   $$
   t \ln(1.5) = \ln(3.375)
   $$

- Solve for $ t $:
   $$
   t = \frac{\ln(3.375)}{\ln(1.5)}
   $$

5. **Calculate the Value of $ t $:**
   - Use a calculator to find the natural logarithms:
   $$
   \ln(3.375) \approx 1.216
   $$
   $$
   \ln(1.5) \approx 0.405
   $$

- Divide the logarithms:
   $$
   t = \frac{1.216}{0.405} \approx 3
   $$

### Conclusion:
- The equation that can be solved to determine the number of months $ t $ is:
  $$
  200 \times (1.5)^t = 675
  $$

- It will take approximately 3 months for the elephant to weigh 675 pounds.

Knowee AI · Accepted Answer