Connected Rates of Change

2 A point is moving along the curve \(y = 2 x + \frac { 5 } { x }\) in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.02 units per second. Find the rate of change of the \(y\)-coordinate when \(x = 1\).

The equation of a curve is \(y = 4\sqrt{x} + \frac{2}{\sqrt{x}}\).

1. Obtain an expression for \(\frac{dy}{dx}\). [3]
2. A point is moving along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of \(0.12\) units per second. Find the rate of change of the \(y\)-coordinate when \(x = 4\). [2]

9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 } { ( 2 x + 1 ) ^ { 2 } }\) and \(P ( 1,5 )\) is a point on the curve.

1. The normal to the curve at \(P\) crosses the \(x\)-axis at \(Q\). Find the coordinates of \(Q\).
2. Find the equation of the curve.
3. A point is moving along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.3 units per second. Find the rate of increase of the \(y\)-coordinate when \(x = 1\).

The driving force \(F\) newtons and velocity \(v\) km s\(^{-1}\) of a car at time \(t\) seconds are related by the equation \(F = \frac{25}{v}\).

1. Find \(\frac{dF}{dv}\). [2]
2. Find \(\frac{dF}{dt}\) when \(v = 50\) and \(\frac{dv}{dt} = 1.5\). [3]

A spherical balloon of radius \(r\) cm has volume \(V\) cm\(^3\), where \(V = \frac{4}{3}\pi r^3\). The balloon is inflated at a constant rate of 10 cm\(^3\) s\(^{-1}\). Find the rate of increase of \(r\) when \(r = 8\). [5]

\includegraphics{figure_2} A vase with a circular cross-section is shown in Figure 2. Water is flowing into the vase. When the depth of the water is \(h\) cm, the volume of water \(V\) cm\(^3\) is given by $$V = 4\pi h(h + 4), \quad 0 \leq h \leq 25$$ Water flows into the vase at a constant rate of \(80\pi\) cm\(^3\)s\(^{-1}\) Find the rate of change of the depth of the water, in cm s\(^{-1}\), when \(h = 6\) \hfill [5]

The area of a circular stain is growing at a rate of \(1 \text{ mm}^2\) per second. Find the rate of increase of its radius at an instant when its radius is \(2\) mm. [5]

A curve has equation \(y = \frac{12}{5 - 2x}\).

1. Find \(\frac{dy}{dx}\). [2]

A point moves along this curve. As the point passes through \(A\), the \(x\)-coordinate is increasing at a rate of 0.15 units per second and the \(y\)-coordinate is increasing at a rate of 0.4 units per second.

1. Find the possible \(x\)-coordinates of \(A\). [4]

The volume \(V\) m³ of a pile of grain of height \(h\) metres is modelled by the equation $$V = 4\sqrt{h^3 + 1} - 4.$$

1. Find \(\frac{dV}{dh}\) when \(h = 2\). [4]

At a certain time, the height of the pile is 2 metres, and grain is being added so that the volume is increasing at a rate of 0.4 m³ per minute.

1. Find the rate at which the height is increasing at this time. [3]

Water is poured into an empty cone at a constant rate of 8 cm³/s After \(t\) seconds the depth of the water in the inverted cone is \(h\) cm, as shown in the diagram below. \includegraphics{figure_8} When the depth of the water in the inverted cone is \(h\) cm, the volume, \(V\) cm³, is given by $$V = \frac{\pi h^3}{12}$$

1. Show that when \(t = 3\) $$\frac{dV}{dh} = 6 \sqrt[3]{6\pi}$$ [4 marks]
2. Hence, find the rate at which the depth is increasing when \(t = 3\) Give your answer to three significant figures. [3 marks]

3 \includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-05_424_529_248_815} The diagram shows a cubical closed container made of a thin elastic material which is filled with water and frozen. During the freezing process the length, \(x \mathrm {~cm}\), of each edge of the container increases at the constant rate of 0.01 cm per minute. The volume of the container at time \(t\) minutes is \(V \mathrm {~cm} ^ { 3 }\). Find the rate of increase of \(V\) when \(x = 20\).

3 A curve has equation \(y = \frac { 1 } { 60 } ( 3 x + 1 ) ^ { 2 }\) and a point is moving along the curve.
Find the \(x\)-coordinate of the point on the curve at which the \(x\) - and \(y\)-coordinates are increasing at the same rate.

A spherical balloon is inflated so that its volume increases at a rate of \(10\text{ cm}^3\) per second. Find the rate of increase of the balloon's surface area when its diameter is 8 cm. [For a sphere of radius \(r\), surface area \(= 4\pi r^2\) and volume \(= \frac{4}{3}\pi r^3\)]. [5]

9 Water is poured into a tank at a constant rate of \(500 \mathrm {~cm} ^ { 3 }\) per second. The depth of water in the tank, \(t\) seconds after filling starts, is \(h \mathrm {~cm}\). When the depth of water in the tank is \(h \mathrm {~cm}\), the volume, \(V \mathrm {~cm} ^ { 3 }\), of water in the tank is given by the formula \(V = \frac { 4 } { 3 } ( 25 + h ) ^ { 3 } - \frac { 62500 } { 3 }\).

1. Find the rate at which \(h\) is increasing at the instant when \(h = 10 \mathrm {~cm}\).
2. At another instant, the rate at which \(h\) is increasing is 0.075 cm per second. Find the value of \(V\) at this instant.

9 A curve has equation \(y = 2 x ^ { \frac { 1 } { 2 } } - 1\).

1. Find the equation of the normal to the curve at the point \(A ( 4,3 )\), giving your answer in the form \(y = m x + c\).
   A point is moving along the curve \(y = 2 x ^ { \frac { 1 } { 2 } } - 1\) in such a way that at \(A\) the rate of increase of the \(x\)-coordinate is \(3 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
2. Find the rate of increase of the \(y\)-coordinate at \(A\).
   At \(A\) the moving point suddenly changes direction and speed, and moves down the normal in such a way that the rate of decrease of the \(y\)-coordinate is constant at \(5 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
3. As the point moves down the normal, find the rate of change of its \(x\)-coordinate.

3. \begin{figure}[h] \end{figure} A tablet is dissolving in water.
The tablet is modelled as a cylinder, shown in Figure 1.
At \(t\) seconds after the tablet is dropped into the water, the radius of the tablet is \(x \mathrm {~mm}\) and the length of the tablet is \(3 x \mathrm {~mm}\). The cross-sectional area of the tablet is decreasing at a constant rate of \(0.5 \mathrm {~mm} ^ { 2 } \mathrm {~s} ^ { - 1 }\)

1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) when \(x = 7\)
2. Find, according to the model, the rate of decrease of the volume of the tablet when \(x = 4\)

8. \begin{figure}[h] \end{figure} The population of a town is being studied. The population \(P\), at time \(t\) years from the start of the study, is assumed to be $$P = \frac { 8000 } { 1 + 7 \mathrm { e } ^ { - k t } } , \quad t \geqslant 0$$ where \(k\) is a positive constant.
The graph of \(P\) against \(t\) is shown in Figure 3. Use the given equation to

1. find the population at the start of the study,
2. find a value for the expected upper limit of the population. Given also that the population reaches 2500 at 3 years from the start of the study,
3. calculate the value of \(k\) to 3 decimal places. Using this value for \(k\),
4. find the population at 10 years from the start of the study, giving your answer to 3 significant figures.
5. Find, using \(\frac { \mathrm { d } P } { \mathrm {~d} t }\), the rate at which the population is growing at 10 years from the start of the study.

10 A triangle ABC is made from two thin rods hinged together at A and a piece of elastic which joins \(B\) and \(C\). \(A B\) is a 30 cm rod and \(A C\) is a 15 cm rod. The angle \(B A C\) is \(\theta\) radians as shown in the diagram. The angle \(\theta\) increases at a rate of 0.1 radians per second.
Determine the rate of change of the length BC when \(\theta = \frac { 1 } { 3 } \pi\).

\includegraphics{figure_1} Figure 1 shows a sketch of a segment \(PQRP\) of a circle with centre \(O\) and radius \(5\) cm. Given that • angle \(PQR\) is \(\theta\) radians • \(\theta\) is increasing, from \(0\) to \(\pi\), at a constant rate of \(0.1\) radians per second • the area of the segment \(PQRP\) is \(A\) cm²

1. show that $$\frac{dA}{d\theta} = K(1 - \cos \theta)$$ where \(K\) is a constant to be found. [2]
2. Find, in cm²s⁻¹, the rate of increase of the area of the segment when \(\theta = \frac{\pi}{3}\) [4]