Question Type 7: Finding the related rates between variable given contextual information on their relation Exercises

A ladder of length $10\,\text{m}$ leans against a wall. Its base slides away from the wall at a rate of $1\,\text{m s}^{-1}$. Find the rate of change of the angle $\theta$ between the ladder and the ground at the instant when the base is $6\,\text{m}$ from the wall.

A $13\text{ m}$ ladder leans against a vertical wall. The angle $\theta$ between the ladder and the horizontal ground is decreasing at a rate of $0.1\text{ rad s}^{-1}$.

Calculate the speed at which the base of the ladder is moving away from the wall at the instant when $\theta = \arccos \left( \frac{5}{13} \right)$.

A $5\text{ m}$ ladder leans against a wall. The top slides down at $1.5\text{ m/s}$. How fast is the base moving away from the wall when the top is $4\text{ m}$ above the ground?

A $20\text{ m}$ ladder leans against a vertical wall. The base of the ladder slides away from the wall at a constant rate of $3\text{ m}\,\text{s}^{-1}$. Let $\varphi$ be the angle between the ladder and the wall.

Find the rate of change of the angle, $\frac{\text{d}\varphi}{\text{d}t}$, when the base of the ladder is $12\text{ m}$ from the wall.

A $15\,\text{m}$ ladder slides down a wall with the base moving away at $3\,\text{m/s}$. Find the rate of change of the angle $\theta$ between the ladder and the wall when the top is $9\,\text{m}$ high.

A $30\text{ m}$ ladder slides with its base moving at $2\text{ m s}^{-1}$. Find the rate of change of $\cot \theta$ (where $\theta$ is the angle with the ground) when the base is $18\text{ m}$ from the wall.

A $25\,\mathrm{m}$ ladder leans against a vertical wall on horizontal ground. The base of the ladder slides away from the wall at a constant rate of $4\,\mathrm{m\,s^{-1}}$.

Find the rate of change of the area of the right triangle formed by the ladder, the wall, and the ground at the instant when the base is $7\,\mathrm{m}$ from the wall.

A $17\text{ m}$ ladder slides against a wall so that the top moves down at a rate of $\frac{dy}{dt}$. Find the rate of change of the angle $\theta$ between the ladder and the ground in terms of $\frac{dy}{dt}$ when the top is $8\text{ m}$ above the ground.

A ladder of length $20\,\text{m}$ slides against a vertical wall. The base of the ladder moves away from the wall at a constant rate of $4\,\text{m}\,\text{s}^{-1}$. Find the rate of change of the perimeter of the triangle formed by the ladder, the wall, and the ground at the instant when the base is $12\,\text{m}$ from the wall.

A ladder of length $13\text{ m}$ slides with its base moving at $2\text{ m/s}$. At what distance of the base from the wall is the speed of the top of the ladder half the speed of the base?

A ladder of length $10\,\text{m}$ leans against a vertical wall. The base of the ladder slides away from the wall at a rate of $2\,\text{m/s}$.

Calculate the rate at which the top of the ladder is sliding down the wall when the base is $6\,\text{m}$ from the wall.

A ladder of length $13\text{ m}$ leans against a wall. The base slides away at $2\text{ m/s}$. Find the rate at which the top is sliding down when the base is $5\text{ m}$ from the wall.