To find the rate of increase of the area of an equilateral triangle, we need to use the relationship between the side length of the triangle and its area. The formula for the area of an equilateral triangle with side length s is:
A=‌s2We are given that the side length
s is expanding at the rate of
√3‌cm∕ sec, or
‌=√3‌cm∕ secWe are asked to find the rate of increase of the area
A when the side length is 12 cm . This means we need to find
‌ when
s=12‌cm.
To find
‌, we differentiate the area formula with respect to time
t :
‌=‌(‌s2)Using the chain rule, we get:
‌=‌⋅2s⋅‌ Simplifying, we obtain:
‌=‌s⋅‌ Substitute the given values for
s and
‌ :
‌=‌⋅12‌cm⋅√3‌cm∕ sec Simplify the expression:
‌‌=‌⋅12⋅√3cm2∕ sec‌‌=‌⋅12cm2∕ sec‌‌=18cm2∕ sec Therefore, the rate of increase of the area of the equilateral triangle when the side length is 12 cm is:
Option A:
18cm2∕ sec