Related Rates Problem

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

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

but how can the base still 20root3

even after the legs decrease ?
i just want to know
thank u for the solution

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.

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.

thank you

To find the rate at which the area of the triangle is decreasing when it becomes equilateral, we need to express the area in terms of the side length.

Let's assume the length of each leg of the isosceles triangle is "x". Since the base of the triangle is given as 20√3 cm, we have two sides of length x and one side of length 20√3.

Using the property of an equilateral triangle, we know that all sides are equal. So, when the triangle becomes equilateral, all sides will be of length x.

First, let's find the area of the isosceles triangle in terms of "x":

Area of the isosceles triangle = (1/2) * base * height
Area = (1/2) * (20√3) * (x/2) (height is the perpendicular bisector of the base)

Simplifying, we get:
Area = 10√3 * (x/2)
Area = 5√3 * x

Now, let's differentiate both sides of the equation with respect to time (t) to find the rate of change of the area:

d(Area)/dt = d(5√3 * x)/dt

Since we are given that the length of the leg decreases at a rate of 3 cm/h, we have dx/dt = -3 cm/h (negative sign because the length is decreasing).

Now, substitute the given value into the equation:

d(Area)/dt = d(5√3 * x)/dt = 5√3 * (-3) = -15√3 cm²/h

Therefore, the rate of decreasing of the area of the triangle when it becomes equilateral is -15√3 cm²/h.