Mathematical Methods · Unit 1 · Functions and relations · Reciprocal functions
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A marine biologist studies the relationship between water depth and light intensity at a tropical reef. The light intensity, $I$ (in lux), is modelled by $I(d) = \frac{850}{d}$ for $d > 0$, where $d$ is the depth below the water surface (in metres). The graph below shows this relationship. Which statement correctly describes how light intensity changes as depth increases?
A rectangular garden bed is to be constructed with an area of \(48\) m². The length of the garden bed is \(x\) metres, and the width is \(w\) metres. (a) Show that the width can be expressed as \(w = \frac{48}{x}\). (1 mark) (b) The total amount of edging material required is the perimeter \(P\) metres. Determine an expression for \(P\) in terms of \(x\) only. (1 mark) (c) Find the value of \(x\) that minimises the perimeter. Express your answer correct to 1 decimal place. (2 marks)
A pharmaceutical company models the concentration C(t) (in mg/L) of a drug in a patient's bloodstream t hours after administration using the reciprocal function: C(t) = 120/(t + 2) + 15, for t ≥ 0 Determine: (a) the concentration of the drug at the time of administration [1 mark] (b) the time at which the concentration falls to 30 mg/L [2 marks] (c) the horizontal asymptote of C(t) and explain what this represents in the context of the model [1 mark]
A manufacturing company models the average cost per unit (in dollars) of producing a particular component using the function \(C(x) = \frac{2400}{x} + 15\), where \(x\) is the number of units produced. Determine the average cost per unit when 80 units are produced.