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Convert the equation [tex]f(t) = 190(1.33)^t[/tex] to the form [tex]f(t) = ae^{kt}[/tex].

- [tex]a =[/tex]
- [tex]k =[/tex]

Give values accurate to three decimal places.

Answer :

Final answer:

To convert f(t)=190(1.33)^t to the form f(t)=ae^kt, we rewrite 1.33 as e^k to directly identify 'k', which is approximately 0.285. The constant 'a' remains as 190.

Explanation:

To convert the equation f(t)=190(1.33)^t into the form f(t)=ae^kt, we need to identify the values of 'a' and 'k'. However, we first need to realize that the base of 'e' in the natural logarithm is hidden in the given equation. Hence, the step we need to apply is to convert the 1.33 in the equation to e.'

In other words, rewrite 1.33 as e^k. This will allow you to identify 'k' directly from the equation. We know from the property of logarithms that:

ln(1.33)=k, or k=ln(1.33). After calculating this, you will find that k = 0.285 approximately.

Now, the constant 'a' remains the same while transforming the equations. Hence, a = 190.

So, the converted equation will look like f(t)=190e^(0.285t).

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Rewritten by : Barada

Answer:

a = 190

k = 0.285

Step-by-step explanation:

f(t) = 190 (1.33)^t

At t = 0, f(t) = 190. So a = 190.

f(t) = 190 e^(kt)

Set equal and solve for k:

190 (1.33)^t = 190 e^(kt)

1.33^t = e^(kt)

ln(1.33^t) = kt

t ln(1.33) = kt

k = ln(1.33)

k = 0.285