Calculus, 2ADV C1 EQ-Bank 12

A block of ice is melting. The mass \(M\) kilograms of the ice block remaining at time \(t\) hours after it begins to melt is given by \(M(t)=50(12-3t)^2, 0 \leqslant t \leqslant 4\).

Find the rate of change of the ice block's mass at any time \(t\). (1 mark)

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How long does it take for the ice block to completely melt? (1 mark)

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At what time is the ice melting at a rate of 2100 kilograms per hour? (2 marks)

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Show Answers Only

a. \(\dfrac{dM}{dt}=-300(12-3t)\)

b. \(4\ \text{hours}\)

c. \(t=\dfrac{5}{3}\ \text{hours}\)

Show Worked Solution

a. \(M(t)=50(12-3t)^2\)

\(\dfrac{dM}{dt}=50 \times 2 \times (-3) \times(12-3t)=-300(12-3t)\)

b. \(\text{Find}\ t\ \text{when}\ \ M(t)=0:\)

\(50(12-3t)^2=0 \ \Rightarrow \ t=4\)

\(\text{Ice block is completely melted at} \ \ t=4 \ \ \text {hours}\)

c. \(\text{Find}\ t \ \text{when}\ \ \dfrac{d M}{d t}=-2100:\)

\(-300(12-3t)\)	\(=-2100\)
\(12-3t\)	\(=7\)
\(-3t\)	\(=-5\)
\(t\)	\(=\dfrac{5}{3}\ \text{hours}\)