Final answer:
To solve the equation 11000 = 44000e⁻⁰.²ᵗ, divide both sides by 44000, take the natural logarithm of both sides, simplify, solve for t, and calculate the approximate value of t.
Step-by-step explanation:
To solve the equation 11000 = 44000e⁻⁰.²ᵗ, we need to isolate the exponential term and then take the natural logarithm of both sides. We start by dividing both sides of the equation by 44000:
11000/44000 = e⁻⁰.²ᵗ
Simplifying the left-hand side gives:
0.25 = e⁻⁰.²ᵗ
To undo the exponential function, we take the natural logarithm (ln) of both sides:
ln(0.25) = ln(e⁻⁰.²ᵗ)
Using the property ln(e⁻⁰.²ᵗ) = -0.²ᵗ, we get:
ln(0.25) = -0.²ᵗ
Finally, we solve for t by isolating the variable:
t = -ln(0.25) / 0.²
Calculating the approximate value of t using a calculator gives:
t ≈ 1.3863 / 0.² ≈ 693.1472
The solution to the equation is t ≈ 693.1472.
Learn more about Solving Exponential Equations