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Integration and Its Applications · Paper 3 — Free Maths AI HL Practice Questions

1ChallengeLAQDefinite integrals and the area under a curve10 marksPaper 3~15 min

A company produces a chemical compound in a batch reactor. The rate of production,

R(t)

, measured in kilograms per hour, is modelled by

R(t) = 5e^{-0.2t} + 2\sin\!\left(\frac{\pi t}{6}\right)

for

0 \leq t \leq 12

, where

t

is the time in hours from the start of a batch.

(a)

Show that the total mass of compound produced during the first 12 hours satisfies

\int_{0}^{12} \left(5e^{-0.2t} + 2\sin\!\left(\frac{\pi t}{6}\right)\right) dt = 25\left(1 - e^{-2.4}\right).

[5 marks]

(b)

Calculate the average rate of production over the 12-hour period. Give your answer in kilograms per hour, correct to 3 significant figures. [2 marks]

(c)

The reactor must be cleaned if the total production in any 12-hour batch falls below 20 kg. Without using a calculator to evaluate

e^{-2.4}

numerically, show algebraically that cleaning is not required for this batch. Hence state the minimum integer value of the threshold, in kilograms, for which the cleaning decision could not be determined without a calculator. [3 marks]

Solutions

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