You CAN Ace Calculus

 derivatives chain rule implicit differentiation basics of related rates precalculus: word problems For related rates problems involving similar triangles, it may help you to review how to set up the ratios. You can find a discussion of this on the similar triangles precalculus page.

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

If you haven't already, we recommend that you read the related rates basics page for information on how to get started on related rates problems.

This page covers related rates problems specifically involving areas. When solving these types of problems, you first draw a picture and pick out the type of geometric figure involved. Once you have a figure with all the distances labeled, you can write down the equations involved.

What To Do With Constants In Related Rates Problems

What do you do with constants that are given in the problem? First of all, you never want to just go in and plug in all your constants before you take the derivative.

Safe Answer - - Wait and plug in your constants only after you have the derivative. So, you would label all distances with variables, take the derivative with respect to t and then plug in all your given constants. This is what you need to do when you first start learning to work related rates problems. After you have some experience, you can go on to the more experienced technique.

Experienced Answer - - Once you learn the basics of related rates problems, you will have a feel for which constants you can plug in right away and which ones you can't. The difference you need to look for is
- if the variable is NOT changing, then you can substitute the constant in before taking the derivative;
- but, if the variable is changing over time, then you must wait until after you take the derivative before you can substitute the constant into the equation.

At this point, it will just confuse you more if we write down a bunch of theory on how to work these problems. You need to actually see one, then work many in order to see patterns. If you haven't already, read the main related rates page. Then, come back here, watch a video or two and try your hand at a few problems.

Circles

When you have a circle and the area of a circle is involved, you need the equation $$A=\pi r^2$$ where $$r$$ is the radius of the circle. Sometimes circumference is involved and that equation is $$C=2\pi r$$.

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. Here is a list converting the old numbering system to the new.

Related Rates - Practice Problems Conversion

[1-1103] - [2-1105] - [3-1106] - [4-1107] - [5-1102]

Please update your notes to this new numbering system. The display of this conversion information is temporary.

GOT IT. THANKS!

A stone is dropped into a pond sending out circular ripples at a rate of 2ft/sec. How fast is the area enclosed by the ripples changing 10 seconds later?

Problem Statement

A stone is dropped into a pond sending out circular ripples at a rate of 2ft/sec. How fast is the area enclosed by the ripples changing 10 seconds later?

$$80\pi ft^2/sec$$

Problem Statement

A stone is dropped into a pond sending out circular ripples at a rate of 2ft/sec. How fast is the area enclosed by the ripples changing 10 seconds later?

Solution

### 2244 solution video

video by Michel vanBiezen

$$80\pi ft^2/sec$$

After dropping a rock in a pond, the ripples form circles where the radius is changing at a rate of 2 meters per second. Calculate the rate of change of the area when the radius is 3 meters.

Problem Statement

After dropping a rock in a pond, the ripples form circles where the radius is changing at a rate of 2 meters per second. Calculate the rate of change of the area when the radius is 3 meters.

Solution

### 1106 solution video

Triangles

When the area of a triangle is involved, your first task is to determine if the triangle is a right triangle. If it is, then the area formula is easy, $$A = bh/2$$. If not, you will need to determine what formula will help you the most, usually one of the law of cosines or law of sines.

A 20ft ladder is leaning against a wall. The foot of the ladder is being pulled away from the wall at a rate of 5 ft/sec while the top of the ladder remains on the wall. What is the rate of change in the area beneath the ladder when the foot is 8ft from the wall?

Problem Statement

A 20ft ladder is leaning against a wall. The foot of the ladder is being pulled away from the wall at a rate of 5 ft/sec while the top of the ladder remains on the wall. What is the rate of change in the area beneath the ladder when the foot is 8ft from the wall?

Solution

### 1107 solution video

video by PatrickJMT

We have a non-right triangle. Side a, which is not changing, has length 10m and side b, also not changing, has length 15m. However, the angle $$\theta$$ between sides a and b is changing at a rate of 0.02rad/sec. How fast is the area of the triangle changing when $$\theta$$ is $$\pi/3$$?

Problem Statement

We have a non-right triangle. Side a, which is not changing, has length 10m and side b, also not changing, has length 15m. However, the angle $$\theta$$ between sides a and b is changing at a rate of 0.02rad/sec. How fast is the area of the triangle changing when $$\theta$$ is $$\pi/3$$?

Hint

$$Area=(1/2)ab\sin(C)$$

Problem Statement

We have a non-right triangle. Side a, which is not changing, has length 10m and side b, also not changing, has length 15m. However, the angle $$\theta$$ between sides a and b is changing at a rate of 0.02rad/sec. How fast is the area of the triangle changing when $$\theta$$ is $$\pi/3$$?

0.7 m2/sec

Problem Statement

We have a non-right triangle. Side a, which is not changing, has length 10m and side b, also not changing, has length 15m. However, the angle $$\theta$$ between sides a and b is changing at a rate of 0.02rad/sec. How fast is the area of the triangle changing when $$\theta$$ is $$\pi/3$$?

Hint

$$Area=(1/2)ab\sin(C)$$

Solution

### 2250 solution video

video by Michel vanBiezen

0.7 m2/sec

Rectangles

Rectangles are probably the easiest of all. The area of a rectangle is $$A=lw$$.

A screen saver displays the outline of a 3 cm by 2 cm rectangle and then expands the rectangle in such a way that the 2 cm side is expanding at the rate of 5 cm/sec and the proportions of the rectangle never change. How fast is the area of the rectangle increasing when the longer side is 12 cm?

Problem Statement

A screen saver displays the outline of a 3 cm by 2 cm rectangle and then expands the rectangle in such a way that the 2 cm side is expanding at the rate of 5 cm/sec and the proportions of the rectangle never change. How fast is the area of the rectangle increasing when the longer side is 12 cm?

120 cm2/sec

Problem Statement

A screen saver displays the outline of a 3 cm by 2 cm rectangle and then expands the rectangle in such a way that the 2 cm side is expanding at the rate of 5 cm/sec and the proportions of the rectangle never change. How fast is the area of the rectangle increasing when the longer side is 12 cm?

Solution

First, we assign variables and get equations for the terms given in the problem statement. It also may help to draw a figure.
Let's label the sides x and y and assign x to the longer side. This means that the side labeled y is expanding at the given rate of $$dy/dt = 5$$ cm/sec.

Secondly, we set up the equations based on what they want us to calculate. They ask us for the rate of change of the area, so we need an area equation. Since this is a rectangle, the area is length times width. In terms of our variables, we have $$A=xy$$. They tell us to determine $$dA/dt$$ when $$x=12$$ cm.

Now we can start to solve this equation. Notice that we can't substitute $$x=12$$ and $$y=8$$ until after we take the derivative since these variables are changing. If we look at the area equation, $$A=xy$$, we have two variables. It is certainly possible to take the derivative at this point, we would need to use the product rule and we would end up with quite a complicated expression. It is easier to reduce this equation to one variable before taking the derivative. To do this, we need to know the relationship between x and y. This information was given in the problem statement saying that the rectangle is initially 3 cm by 2 cm. So we can set up a ratio.
$$\displaystyle{ \frac{3}{x} = \frac{2}{y} }$$
Now we need to determine which variable to solve for. Either one will work but, since we need to eventually solve for $$dA/dt$$ and we are given $$dy/dt$$, it will save time to keep the variable y and get rid of x. So we solve the ratio for x, giving us $$x = 3y/2$$. Substituting this into the area equation, we have
$$\displaystyle{ A = xy = \frac{3y}{2}y = \frac{3y^2}{2} }$$

It is time to take the derivative with respect to t (since this is a related rates problem).
$$\displaystyle{ A = \frac{3y^2}{2} ~~~ \to ~~~ \frac{dA}{dt} = {3y}\frac{dy}{dt} }$$.

Now we can substitute the given values, $$\displaystyle{ \frac{dy}{dt} = 5 }$$. We are given that $$x=12$$ but we don't have any x's in the equation. However, we can use the ratio to find the y value, i.e. $$y = 2x/3 = 2(12)/3 = 8$$. So now we are ready to calculate the answer.

$$\displaystyle{ \frac{dA}{dt} = {3y}\frac{dy}{dt} = 3(8~cm)(5cm/sec) = 120 }$$ cm2/sec