Final answer:
To rewrite y = 4e^(-0.5t) in the form y = a(1+r)^t or y = a(1-r)^t, use the properties of exponents to simplify the exponential term and express it as 4 / a^t = a(1/a)^t. The equation is now in the desired form, with r = -1/a. The percent rate of change is approximately -69.3%.
Step-by-step explanation:
To rewrite y = 4e^(-0.5t) in the form we need to use the properties of exponents to simplify the exponential term.
First, recall that e is Euler's number (approximately 2.71828) and represents the base of the natural logarithm. Using the property e^(-x) = 1/e^x, we can rewrite the given equation as:
y = 4 / e^(0.5t)
Next, we can rewrite e^(0.5t) using the property (a^b)^c = a^(b*c):
y = 4 / (e^0.5)^t
Finally, since e^0.5 is a constant, let's represent it as a. Therefore, the equation can be rewritten as:
y = 4 / a^t = a(1/a)^t
So the equation is now in the form y = a(1+r)^t, where r = -1/a. Rounding to the nearest thousandth, r ≈ -0.693.
To find the percent rate of change, we can multiply r by 100. Therefore, the percent rate of change to the nearest tenth of a percent is approximately -69.3%.