Application of partial derivatives

In summary, the angle theta between the sloping plane ceiling and horizontal floor can be determined using the equilateral triangle method and the equation tan^2(theta) = 4(a^2 + b^2 + c^2 - ab - bc - ac)/3t^2.
  • #1
WY
28
0
Hey,

I have no idea where to start, for this question. I know that I will probably have to use vector and scalar product and use the trig identity tan^2(theta)=sec^2(theta)-1 - and of course partial derivatives

Question:
In order to determine the angle theta which a sloping plane ceiling makes with the horizontal floor, an equilateral traingle of side-length l is drawn on the floor and the height of the ceiling above the three vertices is measured to be a,b and c. Show that:
tan^2(theta) = 4(a^2 + b^2 + c^2 - ab -bc - ac)/3t^2

Thanks in advance for anyone's help!
 
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  • #2
WY said:
Hey,

I have no idea where to start, for this question. I know that I will probably have to use vector and scalar product and use the trig identity tan^2(theta)=sec^2(theta)-1 - and of course partial derivatives

Question:
In order to determine the angle theta which a sloping plane ceiling makes with the horizontal floor, an equilateral traingle of side-length l is drawn on the floor and the height of the ceiling above the three vertices is measured to be a,b and c. Show that:
tan^2(theta) = 4(a^2 + b^2 + c^2 - ab -bc - ac)/3t^2

Thanks in advance for anyone's help!

WY, I've seen your post elsewhere. Others may be like me: I don't know. However, if it were my problem and I HAD to solve it, I tell you what, you can be sure I'd be making a nice size cardboard, wood, plastic whatever sloping ceiling, nothing big, few feet maybe, draw my triangle on the floor, measure the distances and just start working with it. It's an interesting problem and I think, if no one helps you, that's a good place to start. :smile:
 
  • #3
Solution

There is no need of partial derivatives. Here is the solution:

Consider the floor to be the x-y plane and the 3 vertices of the equilateral triangle A(0,0,0), B(t,0,0), C(t/2,(sq. root of 3)t/2,0).

The points in the plane of the ceiling will be
P(0,0,a), Q(t,0,b), R(t/2,(sq. root of 3)t/2,c)

The eqn of the plane of ceiling through these points (after simplificaton) is obtained as
(sq root 3)(a-b)x+(a+b-2c)y+(sq root 3)t(z-a) = 0
Eqn of x-y plane is z = 0

The angle theta between these planes is obtained as
cos(theta) = (sq root 3)t/(sq root of (3(a-b)^2+(a+b-2c)^2+3t^2) )

Using tan^2(theta) = sec^2(theta)-1 and symplifying we get
tan^2(theta) = 4(a^2+b^2+c^2-ab-bc-ca)/3t^2
 

1. What is the definition of a partial derivative?

A partial derivative is a mathematical concept that measures the rate of change of a function with respect to one of its variables, while holding all other variables constant.

2. What are the main applications of partial derivatives?

Partial derivatives are commonly used in fields such as economics, physics, engineering, and statistics to analyze and model complex systems that involve multiple variables. They are also essential in optimization problems and in understanding the behavior of multivariate functions.

3. How are partial derivatives calculated?

To calculate a partial derivative, the variable of interest is treated as the only variable in the function, while all other variables are treated as constants. The derivative is then calculated using the standard rules of differentiation, such as the power rule or the chain rule.

4. Can partial derivatives be used to find maximum and minimum values?

Yes, partial derivatives are commonly used in optimization problems to find the maximum or minimum values of a multivariate function. This is done by setting the partial derivatives equal to zero and solving for the variables.

5. Are there any real-world examples of the application of partial derivatives?

There are many real-world examples of the application of partial derivatives. Some common ones include finding the optimal production levels for a company, determining the optimal portfolio mix for an investment portfolio, and predicting the movement of particles in a fluid using the Navier-Stokes equations.

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