Two sides of a triangle are 8 m and 10 m in length and the angle between them is increasing at a rate of 0.06 rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is π 3 rad.

The area of the triangle is increasing at a rate 1.2 m²/s.

Step-by-step explanation:

Given sides of a triangle are 8 m and 10 m in length.

The angle between the sides of the triangle is increasing at a rate 0.06 rad/s.

It means

If the sides of a triangle are a and b and θ is the angle of the sides.

The area of the triangle is A =

Here, a = 8 m and b = 10 m

Differentiating with respect to t

putting the value of

                =1.2 m²/s

The area of the triangle is increasing at a rate 1.2 m²/s.

3. parallelogram

step-by-step explanation:

we turn each of our sides into direction vectors

ab=b-a=(3 - 7, 1 - 0) = (-4, 1)

bc=c-b=(1,6)

cd=d-c=(4,-1)

da=a-d=(-1,-6)

we see how the opposite sides have negated direction vectors; this is characteristic of a parallelogram.   we have congruent sides ab=cd and bc=da with unequal lengths because 1^2+6^2 is different from (-4)^2+1^2.   so it's not a square or a rhombus.  

we check the dot product of direction vectors of adjacent sides.   we have

ab dot bc = -4(1) + 1(6) = 2

that's not zero so the sides are not perpendicular.   we've ruled out rectangle.

the direction vectors indicate two pairs of congruent parallel sides, so we choose parallelogram over trapezoid.