suppose a triangle has two sides of lengths a and b. if the angle between these sides varies, what is the maximum possible area that the triangle can attain? what can you say about the minimum possible area?

A = (1/2)(ab)sinØ , where Ø is the angle between the two fixed sides a and ab

dA/dØ = (1/2)(ab)cosØ
= 0 for a max of A
then
cosØ = 0
Ø = 90°

the maximum area = (1/2)(ab)sin90° = (1/2)ab
(which of course is the popular formula for the area of a right-angled triangle. )

Common sense will tell us that the minimum area will be obtained when the angle is 0, that is, there is no triangle.

i don't understand where did the d come from and how did you turn it into cos?

To determine the maximum possible area of a triangle with two sides of lengths a and b, we can use the formula for the area of a triangle, which is given by:

Area = (1/2) * a * b * sin(C)

Where C is the angle between the two sides, a and b.

To maximize the area, we must maximize the sin(C) value. The sine function reaches its maximum value of 1 when the angle is 90 degrees (pi/2 radians). Therefore, the maximum possible area occurs when the angle between the sides is 90 degrees. In this case, the formula simplifies to:

Area = (1/2) * a * b * sin(90) = (1/2) * a * b

So, the maximum possible area is equal to half the product of the lengths of the two sides.

As for the minimum possible area, it occurs when the angle between the sides is 0 degrees or when the sides are collinear. In this case, the area of the triangle is 0 since the base of the triangle becomes infinitely small.

Hence, the minimum possible area of the triangle is 0.

To find the maximum possible area of a triangle with two sides of lengths a and b, we can use the formula for the area of a triangle:

Area = (1/2) * a * b * sin(C)

Where C is the angle between the sides of lengths a and b.

To find the maximum area, we need to maximize the value of sin(C). The maximum value of sin(C) is 1, which occurs when C = 90 degrees (or π/2 radians). This means that the maximum possible area of the triangle is:

Maximum Area = (1/2) * a * b * sin(90) = (1/2) * a * b

So, the maximum area of the triangle is equal to half the product of the lengths of the two sides.

On the other hand, to find the minimum possible area, we need to minimize the value of sin(C). The minimum value of sin(C) is 0, which occurs when C = 0 degrees (or 2π radians). This means that when the angle between the sides of lengths a and b is 0 degrees, the triangle would degenerate into a line segment, and its area would be 0.

Therefore, the minimum possible area of the triangle is 0, which occurs when the angle between the sides of lengths a and b is 0 degrees.