# Related Rates problem involving triangle

• biochem850
In summary, a related rates problem involving a triangle is a mathematical problem where the rates of change of different sides or angles of a triangle are connected and need to be solved simultaneously. To solve these problems, one must identify the given information, set up equations using triangle properties and derivatives, use the chain rule to relate rates of change, and solve the resulting equations. These problems have various applications in physics, engineering, and astronomy, but can also present challenges such as correctly identifying variables and knowing which differentiation rules and trigonometric identities to use. Some tips for solving related rates problems involving triangles include drawing diagrams, labeling variables, and practicing with triangle properties and derivatives.
biochem850

## Homework Statement

"At a given instant the legs of a right triangle are 8in. and 6in., respectively. The first leg decreases at 1in/min and the second increases at 2in/min. At what rate is the area increasing after 2 minutes?"

## Homework Equations

A=$\frac{1}{2}$bh

$\frac{db}{dt}$=-1

$\frac{dh}{dt}$=2

## The Attempt at a Solution

A=$\frac{1}{2}$bh

$\frac{dA}{dt}$=$\frac{1}{2}$($\frac{db}{dt}$*h+b*$\frac{dh}{dt}$)

$\frac{dA}{dt}$=$\frac{1}{2}$(-1*10+6*2)

$\frac{dA}{dt}$=$\frac{1}{2}$(2)=1

Therefore the area is increasing at a rate of $\frac{1in^{2}}{min}$ after 2 minutes. Is my reasoning sound (I'm pretty sure my answer is correct but I want to be sure that my work is logical)?

Last edited by a moderator:
I see no problem with this, looks nice!

biochem850 said:

## Homework Statement

"At a given instant the legs of a right triangle are 8in. and 6in., respectively. The first leg decreases at 1in/min and the second increases at 2in/min. At what rate is the area increasing after 2 minutes?"

## Homework Equations

A=$\frac{1}{2}$bh

$\frac{db}{dt}$=-1

$\frac{dh}{dt}$=2

## The Attempt at a Solution

A=$\frac{1}{2}$bh

$\frac{dA}{dt}$=$\frac{1}{2}$($\frac{db}{dt}$*h+b*$\frac{dh}{dt}$)

$\frac{dA}{dt}$=$\frac{1}{2}$(-1*10+6*2)

$\frac{dA}{dt}$=$\frac{1}{2}$(2)=1

Therefore the area is increasing at a rate of $\frac{1in^{2}}{min}$ after 2 minutes. Is my reasoning sound (I'm pretty sure my answer is correct but I want to be sure that my work is logical)?
Your work is logical and mostly correct, but you have a small mistake. The two legs are 8" and 6", not 10" and 6" as you show in your work. The hypotenuse is 10", but it doesn't enter into this problem.

Last edited:
I thought that you're supposed input the length of the two legs after 2 minutes (and according to the derivatives for both legs this would be 6 and 10 after a 2 minute interval)?

Your supposed to input the original lengths?

Sorry, I missed that "after 2 minutes" part in the first post. Your work is fine.

## 1. What is a related rates problem involving a triangle?

A related rates problem involving a triangle is a type of mathematical problem where the rates of change of different sides or angles of a triangle are connected and need to be solved simultaneously. This type of problem involves using the properties of triangles, such as the Pythagorean theorem and trigonometric functions, to find the unknown rates of change.

## 2. How do you approach solving a related rates problem involving a triangle?

To solve a related rates problem involving a triangle, you need to first identify the given information and what needs to be found. Then, you can set up equations using the properties of triangles and their derivatives. Next, you can use the chain rule to relate the rates of change of different variables. Finally, you can solve the resulting equations to find the unknown rates of change.

## 3. What are some common applications of related rates problems involving triangles?

Related rates problems involving triangles are commonly used in fields such as physics, engineering, and astronomy. They can be used to calculate the rates of change of distances, velocities, and angles in various real-life scenarios, such as the motion of objects, the changing size of shadows, and the movement of celestial bodies.

## 4. What are some common challenges when solving related rates problems involving triangles?

One common challenge when solving related rates problems involving triangles is identifying the correct variables and setting up the equations correctly. It is important to carefully read and understand the given information to avoid making mistakes in the setup. Another challenge is knowing which differentiation rules and trigonometric identities to use, which requires a solid understanding of calculus and trigonometry.

## 5. Are there any tips for solving related rates problems involving triangles?

Some tips for solving related rates problems involving triangles include drawing a diagram to visualize the problem, using the given information to set up equations, and labeling all variables clearly. It can also be helpful to review the properties of triangles and their derivatives. Practice and patience are key in mastering this type of problem.

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