Related Rates Problem

Question

An isosceles triangle with a base of 20root3 cm long. If the length of the leg decreases at rate 3 cm/h, find the rate of decreasing of the area of the triangle in the instant at which the triange becomes equilateral

Answer 1

If the leg length is s, the height of the triangle is

h^2 + (10√3)^2 = s^2
or
h^2 + 300 = s^2

The area is
a = 1/2 bh = 10h√3
= 10√3 √(s^2-300)
when the triangle is equilateral, s=20√3

da/dt = 10√3 s/√(s^2-300) ds/dt
= 10√3 * 20√3/√(1200-300) * -3
= -1800/30
= -60 cm^2/hr

Answer 2

but how can the base still 20root3
even after the legs decrease ?
i just want to know
thank u for the solution

Answer 3

Come on - just because you are making the triangle less tall does not mean that the base has to change. It is still isosceles.

The only sticking point comes when the two legs shrink down to 1/2 the base length. At that point the triangle has flattened into a straight line.

Answer 4

Draw the base line AB of the triangle. Now construct the perpendicular bisector of AB.

The vertex C can be anywhere up along that bisector to form an isosceles triangle.

Answer 5

thank you