# Homework Help: Differential problem

1. Nov 29, 2008

### clairez93

1. The problem statement, all variables and given/known data

A side of an equilateral triangle is measured to be 10 cm. Estimate the change in the area of the triangle when the side shrinks to 9.8 cm.

2. Relevant equations

3. The attempt at a solution

$$A = 1/2*bh$$

$$x = 10, dx = 0.2$$

$$b = x/2, h = \sqrt{x^{2} - x^{2}/4}$$

$$1/2(x/2)(\sqrt{3x^{2}/4} = \sqrt{3x^{2}}b / 8$$

$$dy = f'(x)dx = f'(10)(-0.2)$$

$$f'(x) = 8[(3x^{1/2})^{1/2} + 1/2(3x^{2})^{-1/2})(6b)(b)] / 64$$

$$= 8(\sqrt{3x^{2}} + 3x^{2} / \sqrt{3x^{2}) / 64$$

$$= 8(6x^{2} / \sqrt{3x^{2}}) / 64 = 48x^{2}/\sqrt{3x^{2}} / 64 = 48x^{2} / 64\sqrt{3x^{2}}$$

$$dy = (48(10)^{2} / 64\sqrt{3(10)^{2}})(-0.2) = -0.866$$

$$Delta A = f(x + Delta x) - f(x)$$

$$= f(10-0.2) - f(10)$$

$$= f(9.8) - f(10)$$

$$= 20.793 - 21.651$$

$$= -.857$$