Ortiz Turnbull
12/17/2023 · Elementary School

SE 4.1 1. Let \( f(x)=4-x^{2} \). Consider a rectangle with vertices \( (0,0),(x, 0),(x, y) \), and \( (0, y) \). where the vertex \( (x, y) \) lies on the curve \( y=f(x), 0 \leq x \leq 2 \) (see the picture at right). Part I . 1. Express the perimeter \( P \) of the rectangle as a function of \( x, P(x) \). 2. Find the absolute maximum and absolute minimum of \( P(x) \) on the interval \( 0 \leq x \leq 2 \). 3. Find the dimensions of the rectangle with the largest perimeter.

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1. The perimeter \( P(x) \) of the rectangle is \( 8 + 2x - 2x^2 \). 2. The absolute maximum of \( P(x) \) is \( 8 \) at \( x = 0 \) and \( x = 1 \), and the absolute minimum is \( 4 \) at \( x = 2 \). 3. The rectangle with the largest perimeter has dimensions \( 1 \) (width) and \( 3 \) (height).

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