Practical Applications of Connected Rates of Change

The diagram shows the curve $y=2x^2$ and the points $X(-2,0)\text{ and }P(p,0)$. The point $Q$ lies on the curve and $PQ$ is parallel to the $y-$axis.

The point $P$ moves along the $x-$axis at a constant rate of $0.02$ units per second and $Q$ moves along the curve so that $PQ$ remains parallel to the $y-$axis. Find the rate at which area $A$ increasing when $p=2$.[Write your answer excluding units]

The diagram shows the curve $y=2x^2$ and the points $X(-2,0)\text{ and }P(p,0)$. The point $Q$ lies on the curve and $PQ$ is parallel to the $y-$axis. Express area $A$ of triangle $XPQ$ in terms of $p$.

A watermelon is assumed to be spherical in shape while it is growing. Its mass, $M$ kg, and radius, $r$ cm, are related by the formula $M=k{r}^3$, where $k$ is a constant. It is also assumed that the radius is increasing at a constant rate of $0.1$ centimetres per day. On a particular day the radius is $10$ cm and the mass is $3.2$ kg. Find the value of $k$ and the rate at which the mass is increasing on this day.

An oil pipeline under the sea is leaking oil and a circular patch of oil has formed on the surface of the sea. At midday the radius of the patch of oil is $50$ m and is increasing at a rate of $3$ metres per hour. Find the rate at which the area of the oil is increasing at midday.

The volume of a spherical balloon is increasing at a constant rate of $40 {\mathrm{cm}}^3/\mathrm{s}$ per second. Find the rate of increase of the radius of the balloon when the radius is $15$ cm.

The diagram shows a right circular cone with radius $10$ cm and height $30$ cm. The cone is initially completely filled with water. Water leaks out of the cone through a small hole at the vertex at a rate of $4 {\mathrm{cm}}^3/\mathrm{s}$. Find the rate of change of $h$, when $h=20$ cm.

The diagram shows a right circular cone with radius $10$ cm and height $30$ cm. The cone is initially completely filled with water. Water leaks out of the cone through a small hole at the vertex at a rate of $4 {\mathrm{cm}}^3/\mathrm{s}$. Show that the volume of water in the cone, $V {\mathrm{cm}}^3$, when the height of the water is $h$ cm is given by the formula $V=\frac{\pi h^3}{27}$.

A cylindrical container of radius $8$ cm and height $25$ cm is completely filled with water. The water is then poured at a constant rate from the cylinder into an empty inverted cone.The cone has radius $15$ cm and height $24$ cm and its axis is vertical. It takes $40$  seconds for all of the water to be transferred. When the depth of the water in the cone is $10$ cm, Find the rate of change of the horizontal surface area of the water in the cone.

A cylindrical container of radius $8$ cm and height $25$ cm is completely filled with water. The water is then poured at a constant rate from the cylinder into an empty inverted cone.The cone has radius $15$ cm and height $24$ cm and its axis is vertical. It takes $40$  seconds for all of the water to be transferred. When the depth of the water in the cone is $10$ cm, find the rate of change of the height of the water in the cone.

A cylindrical container of radius $8$ cm and height $25$ cm is completely filled with water. The water is then poured at a constant rate from the cylinder into an empty inverted cone.The cone has radius $15$ cm and height $24$ cm and its axis is vertical. It takes $40$  seconds for all of the water to be transferred. If $V$ represents the volume of water, in ${\mathrm{cm}}^3$, in the cone at time $t$ seconds, find $\frac{dV}{dt}$ in terms of $\mathrm{\pi }$.

Paint is poured onto a flat surface and a circular patch is formed. The area of the patch increases at a rate of $5 {\mathrm{cm}}^2/\mathrm{s}$. Find, in terms of $\mathrm{\pi }$, the rate of increase of the radius of the patch after $8$ seconds.

Paint is poured onto a flat surface and a circular patch is formed. The area of the patch increases at a rate of $5 {\mathrm{cm}}^2/\mathrm{s}$. Find, in terms of $\mathrm{\pi }$, the radius of the patch after $8$ seconds.

Oil is poured onto a flat surface and a circular patch is formed. The radius of the patch increases at a rate of $2\sqrt{r} \mathrm{cm}/\mathrm{s}$. Find the rate at which the area is increasing when the circumference is $8\pi \mathrm{cm}$.

Water is poured into the hemispherical bowl of radius $5$ cm at a rate of $3\pi {\mathrm{cm}}^3/\mathrm{s}$. After $t$ seconds, the volume of water in the bowl, $V {\mathrm{cm}}^3$ is given by $V=5\pi h^2-\frac{1}{3}\pi h^3$, where $h$ cm is the height of the water in the bowl. Find the rate of change of $h$ when $h=3$ cm.

Water is poured into the hemispherical bowl of radius $5$ cm at a rate of $3\pi {\mathrm{cm}}^3/\mathrm{s}$. After $t$ seconds, the volume of water in the bowl, $V {\mathrm{cm}}^3$ is given by $V=5\pi h^2-\frac{1}{3}\pi h^3$, where $h$ cm is the height of the water in the bowl. Find the rate of change of $h$ when $h=1$ cm.

The diagram shows a water container in the shape of a triangular prism of length $120$ cm.The vertical cross-section is an equilateral triangle. Water is poured into the container at a rate of $24 {\mathrm{cm}}^3/\mathrm{s}$. Find the rate of change of $h$ when $h=12$ cm.

The diagram shows a water container in the shape of a triangular prism of length $120$ cm.The vertical cross-section is an equilateral triangle. Water is poured into the container at a rate of $24 {\mathrm{cm}}^3/\mathrm{s}$. Show that the volume of water in the container, $V {\text{cm}}^3$, is given by $V=40\sqrt{3}{h}^2$, where $h$ cm is the height of the water in the container.

A closed circular cylinder has radius $r$ cm and surface area $A {\mathrm{cm}}^2$, where $A=2\pi r^2+\frac{400\pi }{r}$. Given that the radius of the cylinder is increasing at a rate of $0.25 \mathrm{cm}/\mathrm{s}$, find the rate of change of $A$ when $r=10$ cm.

A solid metal cuboid has dimensions $x \mathrm{cm}\text{ by }x \mathrm{cm}\text{ by }4x \mathrm{cm}$.The cuboid is heated and the volume increases at a rate of $0.15 {\mathrm{cm}}^3/\mathrm{s}$. Find the rate of increase of $x$ when $x=2$ cm.

A cube has side length $x$ cm and volume $V {\mathrm{cm}}^3$. The volume is increasing at a rate of $1.5 {\mathrm{cm}}^3/\mathrm{s}$. Find the rate of increase of $x$ when $V=8 {\mathrm{cm}}^3$.