Question

the daily dose of ampicillin for the treatment of an ear infection is 115 mg/kg of body weight. what is the daily dose for a 40.-lb child? express your answer to two significant figures.

Explanation:

Step1: Convert pounds to kilograms

We know that \(1\space lb = 0.453592\space kg\), so for a \(40 - lb\) child, the weight in kilograms is \(40\times0.453592\space kg\).
\(40\times0.453592 = 18.14368\space kg\)

Step2: Calculate the daily dose

The daily dose is \(115\space mg/kg\), so for a child with weight \(18.14368\space kg\), the daily dose is \(115\times18.14368\space mg\).
\(115\times18.14368 = 2086.5232\space mg\)

Step3: Round to two significant figures

Rounding \(2086.5232\) to two significant figures, we get \(2100\space mg\) (or \(2.1\times10^{3}\space mg\)). But wait, let's check the significant figures again. Wait, \(40\) has two significant figures (the trailing zero might be ambiguous, but in this context, it's likely two), and \(115\) has three. When multiplying, the result should have two significant figures. Wait, \(40\space lb\) (two sig figs), \(0.453592\) is a conversion factor (exact, so doesn't affect sig figs), \(115\) (three sig figs). So the number of sig figs is determined by \(40\) (two). So \(40\times0.453592 = 18.14368\) (but we consider \(40\) as two sig figs, so \(18\space kg\) (wait, no, \(40\) is two sig figs, so \(40\times0.453592 = 18.14368\), but when we use it in calculation, we should keep more digits during calculation and round at the end. So \(115\times18.14368 = 2086.5232\), now round to two sig figs: \(2100\space mg\) or \(2.1\times10^{3}\space mg\). Wait, but maybe I made a mistake in the conversion. Wait, \(1\space kg = 2.20462\space lb\), so \(40\space lb\div2.20462\space lb/kg\approx18.14\space kg\). Then \(115\space mg/kg\times18.14\space kg = 115\times18.14 = 2086.1\space mg\). Rounding to two significant figures: \(2100\space mg\) (since the first two significant figures are 2 and 1, the next digit is 0, which is less than 5? Wait no, 2086.1, the first two sig figs are 2 and 0? Wait no, 2086.1, the significant figures: 2, 0, 8, 6, 1? No, leading zeros don't count, but here 2 is the first, 0 is the second? Wait no, 2086.1, the number is 2.0861×10³. So two significant figures would be 2.1×10³, which is 2100.

Wait, maybe the conversion factor is \(1\space kg = 2.2\space lb\) (approximate for simplicity). Then \(40\space lb\div2.2\space lb/kg\approx18.18\space kg\). Then \(115\times18.18 = 2090.7\space mg\), rounded to two sig figs is \(2100\space mg\).

Answer:

\(2.1\times10^{3}\) (or \(2100\))