Application of partial derivatives

[WY]

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.

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!

 

[Nate810]

To start, let's draw a diagram of the situation: a b c \ / \ \ / \ \ / \ \ / \ V \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \theta \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \________________\Let's label the sides of the triangle as x, y, and z. Then, we know that the angles opposite each side are equal, and they must all measure theta, so we can say that tan(theta) = x/y = y/z = z/xNow, using the law of cosines, we can write:x^2 = a^2 + b^2 - 2abcos(theta)y^2 = b^2 + c^2 - 2bc cos(theta)z^2 = c^2 + a^2 - 2ac cos(theta)We also know that the sides of an equilateral triangle are equal, so we can set x=y=z=l. Plugging this in to the formulas above, we get:l^2 = a^2 + b^2 - 2abcos(theta)l^2 = b^2 + c^2 - 2bc cos(theta)l^2 = c^2 + a^2 - 2ac cos(theta)Since all three of these equations are equal, we can add them together to get:3l^2 = a^2 + b^2 + c^2 - (ab + bc + ac)cos(theta)We can then use the trig identity tan^2

 

[thepatientmental]

The application of partial derivatives in this problem involves finding the maximum or minimum value of a function with multiple variables. In this case, the function is tan^2(theta), which represents the angle theta of the sloping plane ceiling. The variables are a, b, and c, which represent the height of the ceiling above the three vertices of the equilateral triangle.

To solve this problem, we can use the method of Lagrange multipliers. This involves finding the critical points of the function while considering the constraints given in the problem. In this case, the constraint is that the triangle is equilateral, which means all three sides have the same length.

Using the given information, we can set up the following equations:

tan^2(theta) = (b/a)^2 = (c/a)^2 = (b/c)^2

We can then use the trig identity tan^2(theta) = sec^2(theta) - 1 to rewrite the equation as:

sec^2(theta) - 1 = 4(a^2 + b^2 + c^2 - ab - bc - ac)/3t^2

Next, we can take the partial derivative of both sides with respect to a, b, and c. This will give us a system of equations that we can solve to find the critical points.

After solving for the critical points, we can plug them back into the original equation and determine the maximum or minimum value of tan^2(theta). This will give us the angle theta that the sloping plane ceiling makes with the horizontal floor.

In summary, the application of partial derivatives in this problem allows us to find the maximum or minimum value of a function with multiple variables, which in this case represents the angle theta of the sloping plane ceiling.