The function \( y(t) \) is given by \( y=\mathrm{e}^{t} \) (a) Calculate values of \( t \) such that \( y \) is (i) 1 , (ii) 2 , (iii) 4 , (iv) 8 . (b) Show that if \( t \) is increased by \( \ln 2 \) then the value of \( y \) doubles. (c) A function \( z(t) \) is given by \[ z=\mathrm{e}^{t} \quad k>0 \] Find the increase in the value of \( t \) which will result in the value of \( z \) doublin

To solve the given problems based on the function \( y(t) = \mathrm{e}^{t} \) and later for \( z(t) = \mathrm{e}^{t} \), let's tackle each part step by step. (a) To find values of \( t \) such that \( y \) takes on the values 1, 2, 4, and 8, we can set up the following equations based on our function: 1. For \( y = 1 \): \[ \mathrm{e}^{t} = 1 \quad \Rightarrow \quad t = 0 \] 2. For \( y = 2 \): \[ \mathrm{e}^{t} = 2 \quad \Rightarrow \quad t = \ln(2) \] 3. For \( y = 4 \): \[ \mathrm{e}^{t} = 4 \quad \Rightarrow \quad t = \ln(4) = 2\ln(2) \] 4. For \( y = 8 \): \[ \mathrm{e}^{t} = 8 \quad \Rightarrow \quad t = \ln(8) = 3\ln(2) \] (b) To show that if \( t \) is increased by \( \ln 2 \), the value of \( y \) doubles, we calculate: \[ y(t + \ln(2)) = \mathrm{e}^{t + \ln(2)} = \mathrm{e}^{t} \cdot \mathrm{e}^{\ln(2)} = \mathrm{e}^{t} \cdot 2 = 2y(t) \] Thus, \( y \) doubles when \( t \) increases by \( \ln 2 \). (c) For the function \( z(t) = \mathrm{e}^{t} \), we want to find the increase in \( t \) that results in doubling \( z \). Setting up the equation for doubling: \[ z(t + k) = \mathrm{e}^{t + k} = 2\mathrm{e}^{t} \] This simplifies to: \[ \mathrm{e}^{k} = 2 \quad \Rightarrow \quad k = \ln(2) \] Thus, an increase of \( k = \ln(2) \) in \( t \) will result in \( z \) doubling.