Question

Rewrite f(t) = 0.4 * (1.16) ^ (t - 1) in the form y=a(1+r)^ prime .y=a(1-r)^ prime to determine whether it represents exponential growth or exponential decay Round a and to the nearest hundredth if necessary

Answer 1

y = -0.916 * (1 - 0.140)^t

Explanation:

f(t) = 0.4 * (1.16)^(t - 1)

To rewrite the equation in the form y = a(1 + r)^prime or y = a(1 - r)^prime, we can take the natural logarithm of both sides:

ln(f(t)) = ln(0.4) + (t - 1)ln(1.16)

ln(f(t)) = ln(0.4) + tln(1.16) - ln(1.16)

So,

y = ln(0.4) + tln(1.16)

Now we can rewrite this in the desired form by letting a = ln(0.4) and r = ln(1.16):

y = a + t * r

a = ln(0.4) = -0.916

r = ln(1.16) = 0.140

So,

y = -0.916 + 0.140 * t

y = -0.916 * (1 - 0.140)^t

This form of the equation shows that the function f(t) represents exponential decay. The value of "a" represents the initial value of f(t) and "r" represents the decay rate. Since r is negative, the function decreases over time, which means that the function represents exponential decay.