the sides of an equilateral triangle are increasing at a rate of 27 in/sec. how fast is the triangles area increasing when the sides of the triangle are each 18 inches long?
To find the rate at which the triangle's area is increasing, we can use the formula for the area of an equilateral triangle:
Area = √3/4 * side^2.
First, let's calculate the initial area of the triangle when the sides are 18 inches long:
Initial area, A₁ = √3/4 * (18)^2.
Now, we need to find the derivative of the area with respect to time. Let's denote the rate at which the sides of the triangle are increasing as ds/dt.
To find how fast the area is changing with respect to time, we'll use the chain rule:
dA/dt = dA/ds * ds/dt.
Now, let's differentiate the area formula with respect to the side length:
dA/ds = √3/4 * 2 * side.
Plugging in the side length when the sides are 18 inches long:
dA/ds = √3/4 * 2 * 18.
To find ds/dt, the rate at which the sides of the triangle are increasing, we are given that ds/dt = 27 in/sec.
Now we can substitute the values we've found into the equation:
dA/dt = (√3/4 * 2 * 18) * 27.
Simplifying this expression will give us the rate at which the triangle's area is increasing when the sides are each 18 inches long.