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Calculus, 2ADV C1 EQ-Bank 11

An oil slick on the surface of water forms a circular shape. The radius \(r\) metres of the oil slick is increasing according to the formula  \(r(t)=3 \sqrt{t}\), where \(t\) is the time in minutes after the oil begins to spread,  \(t \geqslant 0\).

  1. Find the rate at which the radius is increasing at any time \(t\).   (1 mark)

  2. At what rate is the area of the oil slick increasing when \(t=16\) minutes?   (1 mark)

Show Answers Only

a.    \(\dfrac{d r}{d t}=\dfrac{3}{2 \sqrt{t}}\)

b.    \(\dfrac{d r}{d t}=\dfrac{3}{8} \ \text{metres/min}\)

Show Worked Solution

a.    \(r(t)=3 \sqrt{t}\)

\(\dfrac{d r}{d t}=\dfrac{1}{2} \times 3 \times t^{-\tfrac{1}{2}}=\dfrac{3}{2 \sqrt{t}}\)
 

b.    \(\text{Find} \ \dfrac{d r}{d t} \ \text{when} \ \ t=16:\)

\(\dfrac{d r}{d t}=\dfrac{3}{2 \times \sqrt{16}}=\dfrac{3}{8} \ \text{metres/min}\)

Calculus, 2ADV C1 EQ-Bank 20

A cylindrical water tank is being filled. The volume \(V\) litres of water in the tank at time \(t\) minutes after filling begins is given by  \(V(t)=500 \sqrt{(2 t+1)}, t \geqslant 0\).

  1. At what rate is water entering the tank at any time \(t\) ?   (1 mark)

  2. Find the rate at which the tank is being filled when \(t=12\) minutes.   (1 mark)

  3. At what time is the water flowing into the tank at a rate of 125 litres per minute?   (2 marks)

Show Answers Only

a.    \(\dfrac{dV}{dt}= \dfrac{500}{\sqrt{2t+1}} \)

b.    \(100\ \text{L/min} \)

c.    \(\dfrac{15}{2}\ \text{mins}\)

Show Worked Solution

a.    \(V(t)=500(2t+1)^{\frac{1}{2}}\)

\(\dfrac{dV}{dt}=\dfrac{1}{2} \times 2 \times 500(2t+1)^{-\frac{1}{2}} = \dfrac{500}{\sqrt{2t+1}} \)
 

b.    \(\text{When}\ \ t=12:\)

\(\dfrac{dV}{dt}= \dfrac{500}{\sqrt{25}} = 100\ \text{L/min} \)
 

c.    \(\text{Find}\ t\ \text{when}\ \dfrac{dV}{dt}=125:\)

\(125\) \(=\dfrac{500}{\sqrt{2t+1}}\)  
\(\sqrt{2t+1}\) \(=4\)  
\(2t+1\) \(=16\)  
\(t\) \(=\dfrac{15}{2}\ \text{mins}\)