# Related rates finding the dA/dt.

• athamz
In summary, the measure of one of the acute angles of a right triangle is decreasing at a rate of 1/36 pi rad/sec. The length of the hypotenuse is constant at 40cm. The area of the triangle can be found using the formula 1/2 base times height, where the base is 40cos(theta) and the height is 40sin(theta). To find the rate of change of the area, differentiate the equation with respect to time and use the chain and product rules. The value of cos(theta) and sin(theta) can be found using the given value of theta. It is possible to get a negative answer, indicating a decrease in area.
athamz

## Homework Statement

The measure of one of the acute angles of a right triangle is decreasing at the rate 1/36 pi rad/sec. If the length of the hypotenuse is constant at 40cm, find how fast the area is changing when the measure of the acute angle is 1/6 pi.

## The Attempt at a Solution

I used the formula for the area of a triangle which is 1/2 base times the height, using trigonometry function, I substitute the value of base with 40costeta and the height is 40sinteta. How will I differentiated A = 40costeta*40sinteta? Will I use the product rule to differentiate? where will I include or put the rate given?

so you have
$$A(\theta) = 40 cos(\theta) sin(\theta)$$

differentiate both sides with respect to t, you will need to use both chain & product rules

oh.. meaning

$$dA/dt = (cos^2 [/theta] - sin^2 [/theta]) d[/theta]/dt$$

Where will I get the value for cos[/theta] and sin[/theta] ?

well you're given theta

oh.. thank you so much..
Is it alright if I will get a negative answer?

A note: This equation actually becomes very easy to differentiate if you recognize that $40 sin(\theta) cos(\theta) = 20 sin(2 \theta)$.

i haven't checked if it is the correct answer, but assuming you've done the maths correctly a negative answer is fine, it just means the area is decreasing

## 1. What is the formula for related rates?

The formula for related rates is dA/dt = dA/dx * dx/dt, where dA/dt represents the rate of change of the area, dA/dx represents the rate of change of the area with respect to x, and dx/dt represents the rate of change of x.

## 2. How do you find the rate of change of area (dA/dt) in related rates problems?

To find dA/dt, you need to differentiate the formula for the area with respect to time (t) using the chain rule. Then, plug in the given values and solve for dA/dt.

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

Related rates problems are commonly used in physics and engineering to analyze the changing rates of various quantities, such as the volume of a gas, the height of a falling object, or the area of a growing circle.

## 4. How do you choose which variables to differentiate in related rates problems?

In related rates problems, you need to identify which quantities are changing and how they are related to each other. Then, choose the appropriate variables to differentiate based on the given information and the formula for the related rates.

## 5. Can you provide an example of a related rates problem?

Sure, here's an example: A cylindrical tank is being filled with water at a rate of 2 cubic feet per minute. If the tank has a height of 10 feet and a radius of 5 feet, at what rate is the water level rising when the water is 6 feet deep? In this problem, the rate of change of the water level (dA/dt) can be found by differentiating the formula for the volume of a cylinder with respect to time (t). Then, substitute the given values and solve for dA/dt to find that the water level is rising at a rate of 0.2 feet per minute.

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