Two sides of a triangle are 4 inches long. What should be the angle between these sides to make the area of the triangle as large as possible?

Since you are in Calculus, I assume you know the area of a triangle is (1/2) ab sinØ, where Ø is the contained angle between a and b

A = (1/2)(4)(4)sinØ
A = 8sinØ
dA/dØ = 8cosØ
for a max of A, 8cosØ = 0
Ø = 90° or 270°
but we are in a triangle so the angle must be 90° for the largest area

Not what I expected, but quite true. The equilateral triangle has area 4√3 < 8

To find the angle that will maximize the area of the triangle, we can use the fact that the area of a triangle is given by the formula:

Area = (1/2) * base * height

In this case, the "base" of the triangle can be one of the sides, and the "height" will be the perpendicular distance between the base and the third side.

To maximize the area, we need to maximize the height. The height will be largest when it is perpendicular to the base. Therefore, to maximize the area, we need to find the angle between the given sides that will make the third side perpendicular to one of them.

In a triangle, if one side is perpendicular to another, we have a right triangle. The angle between the two sides can be found using trigonometry.

Let's call the two given sides, which are each 4 inches long, side A and side B. The third side, which we want to be perpendicular to one of the given sides, we'll call it side C.

To determine the angle between sides A and B, we can use the trigonometric function tangent (tan).

The tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side.

In this case, for a right triangle, one of the given sides will be adjacent to the angle, and the other side will be opposite.

Let's assume side A is adjacent to the angle, and side B is opposite.

Then, we have:

tan(angle) = B / A

Plugging in the values:

tan(angle) = 4 / 4

Simplifying:

tan(angle) = 1

To find the angle, we can take the inverse tangent (arctan) of both sides:

angle = arctan(1)

Using a calculator, we find:

angle ≈ 45 degrees (or π/4 radians)

Therefore, the angle between the two sides, to make the area of the triangle as large as possible, should be approximately 45 degrees.

do some setup and I'm sure you will find that the problem is solved by a regular n-gon regardless of the number of sides.

Just define the third side in terms of the included angle.