O período de duplicação é o tempo necessário para que uma grandeza que cresce exponencialmente dobrar seu valor. Calcule o período de duplicação de preços que estão subindo a uma taxa de 5 % ao ano.

Passo 1

Vamos entender o que a questão está pedindo. A gente precisa calcular o período de duplicação, que é o tempo que leva para os preços dobrarem, sabendo que eles estão crescendo a uma taxa de 5% ao ano. Para resolver isso, vamos usar a matemática do crescimento exponencial.

Passo 2

Primeiro, vamos lembrar a fórmula para calcular o período de duplicação em crescimento exponencial. A fórmula é:
Aqui, é o tempo para duplicar, é o logaritmo natural de 2, e é a taxa de crescimento (em decimal).

Passo 3

No nosso caso, a taxa de crescimento é de 5% ao ano. Para usar na fórmula, a gente precisa converter esse percentual para decimal:

Passo 4

Agora, vamos substituir os valores na fórmula:
Calculando , que é aproximadamente 0,693:

Passo 5

Fazendo a conta:

Resposta

O período de duplicação dos preços é de aproximadamente 13,86 anos.

Exercícios relacionados

No exercício a seguir, exprese e como funções de u e v utilizando a regra da cadeia, em seguida calcule e no ponto (u, v) dado.
Calcula as derivadas parciais de primeira ordem e para a função:
No exercício a seguir, exprese e como funções de u e v utilizando a regra da cadeia, em seguida calcule e no ponto (u, v) dado.
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