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(a) Let B=5750(1.075)^t give the balance of a bank account after t years . If the formula for B is written B=5750e^kt estimate the value of k correct to four decimal places. (b) The balance of a bank account after t years is given by the formula B=7250e^0.075t . If the formula for B is written B=7250b^t find b exactly , and give the value of b correct to four decimal places. b=e^0.075 bapprox (to 4 decimal places)

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James

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In order to solve the problems for (a) and (b), we're required to use the properties of exponential functions and logarithms. (a) We're given that B=5750(1.075)^{t}. In order to rewrite the formula in the form B=5750e^{kt}, we need  e  kt\text { e }^{ \text { kt}} = 1.075^{t}. Let's take natural logarithms on both sides: \begin{aligned} &\ln(e^{kt})=\ln((1.075)^{t})\\ &k\cdot t Ln(e) = t \cdot Ln(1.075) \\ &kVentuallyennai calculuswithin \end{aligned} Therefore, \boxed{k \approx 0.0715}.(b) You're given that B=7250e^{0.075t}. If you rewrite the formula in the form B=7250b^{t}, this involves transforming b=e^{0.075}. Let's find the value of b; b=\begin{aligned} &e^{0.075} \end{aligned} Therefore, \boxed{b\approx 1.0776}.

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