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clairez93
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Homework Statement
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.
Homework Equations
The Attempt at a Solution
[tex] A = 1/2*bh [/tex]
[tex] x = 10, dx = 0.2[/tex]
[tex]b = x/2, h = \sqrt{x^{2} - x^{2}/4}[/tex]
[tex]1/2(x/2)(\sqrt{3x^{2}/4} = \sqrt{3x^{2}}b / 8 [/tex]
[tex]dy = f'(x)dx = f'(10)(-0.2)[/tex]
[tex]f'(x) = 8[(3x^{1/2})^{1/2} + 1/2(3x^{2})^{-1/2})(6b)(b)] / 64[/tex]
[tex] = 8(\sqrt{3x^{2}} + 3x^{2} / \sqrt{3x^{2}) / 64[/tex]
[tex] = 8(6x^{2} / \sqrt{3x^{2}}) / 64 = 48x^{2}/\sqrt{3x^{2}} / 64 = 48x^{2} / 64\sqrt{3x^{2}}[/tex]
[tex]dy = (48(10)^{2} / 64\sqrt{3(10)^{2}})(-0.2) = -0.866[/tex]
[tex]Delta A = f(x + Delta x) - f(x)[/tex]
[tex]= f(10-0.2) - f(10)[/tex]
[tex]= f(9.8) - f(10)[/tex]
[tex]= 20.793 - 21.651[/tex]
[tex]= -.857[/tex]
Answer Key: -1.732