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Fashion The function \(K(x)=1.3 x-4.7\) converts a men's shoe size in the United States to the equivalent shoe size in the United Kingdom. Determine the function \(K^{-1}(x)\) that can be used to convert a United Kingdom men's shoe size to its equivalent U.S. shoe size.

Escherichia coli (E. coli) is a bacterium that can reproduce at an exponential rate. The \(E\) coli reproduce by dividing. A small number of E. coli bacteria in the large intestine of a human can trigger a serious infection within a few hours. Consider a particular E. coli infection that starts with \(100 E .\) coli bacteria. Each bacterium splits into two parts every half hour. Assuming none of the bacteria die, the size of the \(E .\) coli population after \(t\) hours is given by \(P(t)=100 \cdot 2^{2 t},\) where \(0 \leq t \leq 16\) a. Find \(P(3)\) and \(P(6)\) b. Use a graphing utility to find the time, to the nearest tenth of an hour, it takes for the \(E .\) coli population to number 1 billion.

Lead shielding is used to contain radiation. The percentage of a certain radiation that can penetrate \(x\) millimeters of lead shielding is given by \(I(x)=100 e^{-1.5 x}\) a. What percentage of radiation, to the nearest tenth of a percent, will penetrate a lead shield that is 1 millimeter thick? b. How many millimeters of lead shielding are required so that less than \(0.05 \%\) of the radiation penetrates the shielding? Round to the nearest millimeter.

Graph \(f(x)=e^{x},\) and then sketch the graph of \(f\) reflected across the line given by \(y=x\)

Use Boyd's formula to estimate the body surface area of a patient with the given weight and height. Round to the nearest hundredth of a square meter. \(W=180\) pounds \((81,646.6 \text { grams) } ; H=6 \text { feet } 1\) inch \((185.42\) centimeters)