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Math · Calculus
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A microprocessor chip is being designed with a given rectangular area A. Show that the chip with the minimum perimeter should be a square. Let the length and width of the chip be x and y, respectively. An equation involving the three variables A, x, and y is (Type an equation.) If P is the perimeter of the rectangle, then an equation involving the three variables P, x, and y is (Type an equation.) Use the equation involving A to rewrite the equation involving P so that it uses only P and x. The result describes P as a function of the single variable x. (Type an equation.) dP dx Find dP dx Set the derivative equal to zero, and solve the resulting equation to find the value for x for which a minimum may occur. What is the result? Note that because x is the length of the chip, it must be positive. To check that this x-value does indeed make P a minimum, use the second derivative. Find「 dx dx Click to select your answer(s)A microprocessor chip is being designed with a given rectangular area A. Show that the chip with the minimum perimeter should be a square. Set the derivative equal to zero, and solve the resulting equation to find the value for x for which a minimum may occur. What is the result? Note that because x is the length of the chip, it must be positive. To check that this x-value does indeed make P a minimum, use the second derivative. Find d P dx2 At the x-value found earlieris ▼| This means P | ▼| a minimum at that x-value. dx Find the y-value that corresponds to the x-value found earlier. Why does it follow that the chip with the minimum perimeter should be a square? O A. O B. Since x- 2y, the length of the chip is exactly twice its width. Substituting x and y into the formula for the area confirms that the area of the chip is A. Since y = x, the sides of the rectangle have the same length. Substituting x and y into the formula or the perimeter yields P= 0 Since the perimeter cannot be negative, this must be the minimum perimeter C. O D Click to select your answer(s).

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