the side of an equilateral triangle decreases at a rate of 3cm/s. at what rate is the area decreasing when the area is 150cm squared?
altitude = (s/2)sqrt3
A = (s/2) (s/2) sqrt 3 = (s^2/4) sqrt3
dA/dt = (2 s/4) sqrt3 ds/dt
= (s/2) sqrt 3 * (-3)
= (-3s/2) sqrt 3
now what is s when A = 150?
(s^2/4) sqrt 3 = 150
s^2 = 600/sqrt3
so
dA/dt = (-3/2) sqrt3 (sqrt 600)/3^.25
= (-3/2)(sqrt600)(3^.25)
To determine at what rate the area of an equilateral triangle is decreasing when the area is 150 cm², we will use the formula for the area of an equilateral triangle and apply the concept of related rates.
The formula for the area of an equilateral triangle is given by: A = (√3/4) * s², where A represents the area and s represents the length of the side of the triangle.
We are given that the side of the equilateral triangle is decreasing at a rate of 3 cm/s, which means ds/dt = -3 cm/s (negative because the side length is decreasing). We need to find dA/dt, the rate at which the area is changing.
Differentiate both sides of the area formula with respect to time (t) to find the relationship between dA/dt and ds/dt:
dA/dt = (√3/4) * 2s * ds/dt
Now substitute the values provided:
ds/dt = -3 cm/s (given)
A = 150 cm² (given)
We can rearrange the equation and substitute the given values to find dA/dt:
dA/dt = (√3/4) * 2s * ds/dt
dA/dt = (√3/4) * 2(√(4A/√3)) * (-3) [Substituting s = √(4A/√3) and ds/dt = -3]
dA/dt = (-3√3/√3) * (√(4A/√3))
dA/dt = -6√3 * (√(4A/√3))
dA/dt = -6√3 * √(4A/√3)
dA/dt = -6√(12A)
Therefore, when the area is 150 cm², the rate at which the area is decreasing is -6√(12 * 150) cm²/s.