Answer to Question #281955 in Calculus for Bret

Question

The sides of an equilateral triangle are increasing at the rate of 3 cm/min. Find: a) the rate of change of the perimeter.

b) the rate of change of the area when the side is 3 cm. long.

Answer 1

Solution;

(a)

Perimeter of the triangle;

$P=3a$

Where a is the length of one side.

Hence;

$\frac{dP}{dt}=\frac{d(3a)}{dt}$

$\frac{dP}{dt}=3\frac{da}{dt}$

$\frac{dP}{dt}=3×3cm/min$

Rate of change of perimeter is;

$9cm/min$

(b)

Area of an equilateral triangle is;

$A=\frac{\sqrt3}{4}a^2$

Differentiate with respect to t;

$\frac{dA}{dt}=\frac{\sqrt4}{3}(2a)\frac{da}{dt}$

Substitute a=3cm and $\frac{da}{dt}=3cm/min$

Hence;

$\frac{dA}{dt}=\frac{\sqrt3}{4}×2×3×3cm/min$

The rate of change of are is;

$4.5\sqrt3cm^2/min$