11.22 Related rates of change
Let y = \left(x + 5\right)^{5} be defined as a composition of the functions y = u^{5} and u = x + 5.
Find:
The volume V of oxygen in a scuba diver’s oxygen cylinder is given by V = \dfrac{22}{P}, where P is the pressure inside the tank.
Find the rate of change of V with respect to P.
During a dive, the pressure P inside the cylinder increases at 0.5 units per second. Find the rate of change of the volume of oxygen when P = 2. Let t represent time in seconds.
A spherical hot air balloon, whose volume and radius at time t are V \text{ m}^3and r \text{ m} respectively, is filled with air at a rate of 4 \text{ m}^3 /\text{min}.
At what rate is the radius of the balloon increasing when the radius is 2 \text{ m} ?
A point moves along the curve y = 5 x^{3} in such a way that the x-coordinate of the point increases by \dfrac{1}{5} units per second. Let t be the time at which the point reaches \left(x, y\right).
Find the rate at which the y-coordinate is changing with respect to time when x = 9.
A giant ice sculpture is constructed in the shape of a cube with sides measuring 4 metres. The sculpture is melted in such a way that it always retains the shape of a cube until it is completely melted.
Let V represent the volume of ice remaining in centimetres cubed, and x represent the side length of the ice cube in centimetres at time t seconds after it starts being heated.
If the length of each edge is decreasing by 0.03 \text{ cm/s} , find the initial rate at which the volume of the cube is changing, in 0.03 \text{ cm}^3/\text{s} .
If the volume is decreasing at a rate of 500 \text{ cm}^3/\text{s}, find the initial rate at which the length of each edge is changing in centimetres per second ( \text{ cm/s} ).
A magical water slide has a varying slope given by \dfrac{d y}{d x} = - \tan a, where a is the angle formed by the tangent at a point and the ground. Hermione goes down the magical water slide, where her horizontal velocity is {\dfrac{d x}{d t} = 50 t \cos a}.
Find Hermione's vertical velocity when:
a = \dfrac{\pi}{6} and t = 7
a = \dfrac{\pi}{6} and t = 26
At a theme park, a roller coaster is being designed onto a fixed ring 49 metres above the ground. The angle from the start of the ride to a point on the track anticlockwise is a, and 0 \leq a \leq 2 \pi. The angular velocity of a car on the roller coaster is \dfrac{d a}{d t} = k \left(1 - \cos \left(a\right)\right). The vertical displacement of the roller coaster from the ground at an angle a is {y = 49 + 40 \sin \left(a\right)}.
Use the chain rule to form an expression for the vertical velocity, v.
Find the values of a for all the stationary points of v.
Fill in the table of values to classify the stationary points.
| a | 0 | \dfrac{\pi}{3} | \pi | \dfrac{5\pi}{3} | 2\pi |
|---|---|---|---|---|---|
| v |
Safety regulations require that the vertical velocity of the rollercoaster never exceeds 71 \text{ m/min}. What values of k satisfy this requirement?
A colony of bacteria is being studied on a petri dish. The number of bacteria is n = 2 e^{ 9 t}, where t is the time in minutes since the bacteria was placed onto the dish. The rate at which the area, A, of the dish is covered by the bacteria is \dfrac{d A}{d t} = 5 \text{ mm}^2/\text{min}.
Find the rate, at which the number of bacteria grows relative to the area the bacteria covers, \dfrac{dn}{dA}, for the following values:
t = 10. Write your answers in standard form to three significant figures.
Physicists want to observe the thermal expansion of a particular material in order to find its thermal expansion coefficient, \alpha. The material is shaped into a sphere and placed in the sun for observations. The rate of change of the atmospheric temperature, T, with respect to the volume, V, is given by \dfrac{d T}{d V} = \dfrac{1}{3} \alpha V.
The temperature over the course of the day is given by T \left( t \right) = 5 t \left(24 - t\right) + 12, where t is the number of hours that have passed. Find the maximum temperature during the course of the day.
Find the rate of change of the volume in terms of V when t = 2.
When T = 2 the surface area, S, is 81 \pi \text{ cm}^2 and \dfrac{d V}{d t} = -\dfrac{1}{8} \text{ cm}^3/ \text{s}. Using the formula for the volume of a sphere with respect to its surface area, V = \dfrac{\sqrt{S^{3} }}{6 \sqrt{\pi}}, find \alpha.
A sandbox is being filled at a local playground. The density of sand is given by \\ \dfrac{d m}{d V} = 4.5 \text{ g} /\text{cm}^{3} , where m is the mass in grams and V is the volume in cubic centimetres.
If sand is being added to the sandbox at a rate of \dfrac{d V}{d t} = 400 \text{ cm}^3/\text{min}, find \dfrac{d m}{d t}, the rate at which the sandbox gets heavier over time.
The angle of elevation of a 4 \text{ m} tall tree from the end of its shadow is a. The angular velocity of the angle of elevation as the sun moves across the sky is \dfrac{\pi}{8} radians per hour. Let the length of the shadow be x \text{ m} and the length of the hypotenuse be y \text{ m}.
Find the rate of change of x when {a = \dfrac{\pi}{6}}.
Find the rate of change of y when
a = \dfrac{\pi}{6}.
Find the value of \dfrac{d y}{d x} when a = \dfrac{\pi}{6}.
A water solution is poured into a cone-shaped container at a rate of 4 \text{ cm}^3/\text{sec} as shown in the diagram. h represents the height of the water, r represents the radius of the surface of the water solution, and V is the volume of water in the container at time t seconds after we start pouring the solution.
Show that the volume of solution in the cone is given by the equation \\ V = \dfrac{1}{3} \pi h^{3} \tan ^{2}\left(17 \degree\right).
Find the rate at which the height of water is increasing over time when the height is 7 \text{ cm}. Round your answer to three significant figures.
A takeaway coffee cup is formed by slicing a cone h centimetres below the tip and turning it upside down.
If the base of the cone has a radius 5 \text{ cm}and the total height of the cone is 15 \text{ cm}, show that the volume of the coffee cup can be given by {V = \dfrac{25 \pi}{3} \left(15 - \dfrac{h^{3}}{225}\right) \text{ cm}^3}.
Find \dfrac{d V}{d h}.
Consider a coffee cup that is formed by slicing this cone 6 \text{ cm} below the tip when turned upside down. If the height of the coffee in this coffee cup is increasing at a constant rate of 3 \text{ cm/sec}, find the rate at which the coffee cup is being filled per second when the height of coffee in the cup is 6\text{ cm}. Round your answer to one decimal place.
In a novel about a heroine’s quest, the heroine has found themselves trapped in a room, littered with debris and waste. To her surprise, the two walls x \text{ m} apart begin to cave in. To stop the walls from crushing her, she wedges a metal pole she found from the debris into the corners of the room, as shown in the image. The metal pole forms an angle of \theta with the floor. Yet in spite of her efforts the wall continues to cave in.
In order to find how fast the walls are caving in, she first needs to calculate an expression for x in terms of \theta.
Show that: x = \sqrt{\dfrac{25}{\tan ^{2}\left(\theta\right)} - 225}.
She notices that the rate at which the angle theta changes over time is \dfrac{1}{25} radians/min. Find the rate at which the walls are caving in over time, \dfrac{dx}{dt}, in terms of \theta.