Solutions Homework 9

Solutions Homework 9 - MATH 106 CALCULUS I FOR BIO. &...

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Unformatted text preview: MATH 106 CALCULUS I FOR BIO. & SOC. SCI. FALL 2011 INSTRUCTOR: JOS ´ E MANUEL G ´ OMEZ Solutions Homework 9. Please show all your work. (1) (10 Points) What is the largest area for a right angle triangle whose hypotenuse is 10 cm long? Solution: Let’s denote the sides of the triangle by x and y . The area of the triangle is A = xy 2 . We want to maximize the function A . Using the Pythagorean theorem we get x 2 + y 2 = 100. Solving for y we obtain y = ± √ 100 − x 2 . Note that y is a length so it is positive. We conclude then that y = √ 100 − x 2 . It follows that A ( x ) = x √ 100 − x 2 2 . Let’s determine next the domain of A . To do so note that x is a length so that x ≥ 0. On the other hand, since the hypotenuse of the triangle is 10 we get x ≤ 10. Thus we want to maximize the function A on the domain [0 , 10]. Let’s find next the critical points. For this we have A ′ ( x ) = 1 2 parenleftbigg √ 100 − x 2 + − x 2 √ 100 − x 2 parenrightbigg = 1 2 parenleftbigg 100 − 2 x 2 √ 100 − x 2 parenrightbigg Note that A ′ ( x ) = 0 if and only if 1 2 parenleftbigg 100 − 2 x 2 √ 100 − x 2 parenrightbigg = 0 and thus 100 − 2 x 2 = 0. It follows that x 2 = 50 hence x = ± √ 50. Since the domain of A is the interval [0 , 10] we get that √ 50 is a critical point. Also note that A ′ ( x ) does not exist when...
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This note was uploaded on 01/20/2012 for the course CALC AS.110.106 taught by Professor Josegomez during the Fall '11 term at Johns Hopkins.

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Solutions Homework 9 - MATH 106 CALCULUS I FOR BIO. &...

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