Home / IB Mathematics SL 5.1 Derivative interpreted as gradient function AI SL Paper 1- Exam Style Questions

IB Mathematics SL 5.1 Derivative interpreted as gradient function AI SL Paper 1- Exam Style Questions- New Syllabus

IB DP Maths AI SL Paper 1 - All Topics

Question

Marco charges a client per hour to rent his boat. It is known that
\[ \frac{dP}{dt}=20-\frac{980}{t^{2}}, \qquad 0<t\le 12 \]
where \(P\) is the cost per hour, in Norwegian krone (NOK), that the client is charged and \(t\) is the time, in hours, spent aboard the boat.
The cost per hour has a local minimum when the boat is rented for \(h\) hours.
(a) Find the value of \(h\). [2]
Layla hired Marco’s boat for \(5\) hours and was charged NOK \(328\) per hour. Priya hires Marco’s boat for \(7\) hours.
(b) Show that the cost per hour for Priya is NOK \(312\). [6]
▶️Answer/Explanation
Markscheme

(a)

A local minimum occurs where \(\dfrac{dP}{dt}=0\) (within the given domain). Solve \[ 0=20-\frac{980}{t^2} \ \Longrightarrow\ \frac{980}{t^2}=20 \ \Longrightarrow\ t^2=\frac{980}{20}=49 \ \Longrightarrow\ t=\sqrt{49}=7\ \text{hours}. \] Hence \(\boxed{h=7}\). M1 A1

[2 marks]

(b)

Recognize we must integrate \(\dfrac{dP}{dt}\) to obtain \(P(t)\): \[ P(t)=\int\!\left(20-\frac{980}{t^2}\right)\!dt =20t+\int-980\,t^{-2}\,dt =20t+980\,t^{-1}+C =20t+\frac{980}{t}+C. \] Use Layla’s information \(t=5,\ P=328\) to find \(C\): \[ 328=20(5)+\frac{980}{5}+C =100+196+C \ \Rightarrow\ C=328-296=\boxed{32}. \] Thus \[ P(t)=20t+\frac{980}{t}+32. \] For Priya, \(t=7\): \[ P(7)=20(7)+\frac{980}{7}+32 =140+140+32 =\boxed{312\ \text{NOK}}. \] M1 A1 A1 M1 A1 AG

[6 marks]

Total Marks: 8
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