To solve the problem, we need to find the number of months, [tex]\( t \)[/tex], it will take for the elephant to gain weight from 200 pounds to 675 pounds given that each month the elephant's weight increases exponentially by half the previous month's weight.
### Step-by-Step Solution:
1. Understanding Exponential Growth: The weight of the elephant grows exponentially:
[tex]\[
W_{t} = W_0 \left(\frac{3}{2}\right)^t
\][/tex]
where [tex]\( W_0 \)[/tex] is the initial weight, [tex]\( t \)[/tex] is the number of months, and [tex]\( \left(\frac{3}{2}\right)^t \)[/tex] represents the factor by which the weight increases each month.
2. Setting up the Equation: We know the initial weight [tex]\( W_0 = 200 \)[/tex] pounds, and we need to find the time [tex]\( t \)[/tex] when the weight [tex]\( W_t \)[/tex] reaches 675 pounds:
[tex]\[
200 \left(\frac{3}{2}\right)^t = 675
\][/tex]
3. Solving the Equation:
[tex]\[
200 \left(\frac{3}{2}\right)^t = 675
\][/tex]
By dividing both sides by 200:
[tex]\[
\left(\frac{3}{2}\right)^t = \frac{675}{200}
\][/tex]
Simplifying the fraction on the right-hand side:
[tex]\[
\left(\frac{3}{2}\right)^t = 3.375
\][/tex]
4. Finding the Solution:
We solve for [tex]\( t \)[/tex] to determine the number of months:
[tex]\[
t = 3
\][/tex]
Thus, the correct equation representing the exponential growth is [tex]\( 200\left(\frac{3}{2}\right)^t = 675 \)[/tex], and the number of months it will take for the elephant's weight to reach 675 pounds is 3 months.
Hence, the correct selections from the tables given are:
Equation:
[tex]\[
200\left(\frac{3}{2}\right)^t = 675
\][/tex]
Solution:
[tex]\[
3 \text{ months}
\][/tex]
