Need answers to these problems! Thank you Find the area of the largest rectangle

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Need answers to these problems! Thank you

Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3 cm and 8 cm if two sides of the rectangle lie along the legs. cm2 6. -1.25 points My Notes The top and bottom margins of a poster are each 6 cm and the side margins are each 4 cm. The area of printed material on the poster is fixed at 384 cm. Find the dimensions of the printed area that minimize the area of the whole poster width height cm cm 7. ÷ -11.25 points SCalcET847.504.XP My Notes Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola. y-5-x2 width units height units 8, 1.25 points Find the point on the line 3x + y = 7 that is closest to the point (-5,3). (x, y) = SCalcET8 4.7. 505.xP My Notes ÷

Explanation / Answer

Note first that the formula we would like to maximize is A = (8 x)(y). It remains, then, to eliminate either x or y from the equation (we need x in terms of y or vice versa). To do so, note that the large triangle is similar to the triangle of the rectangle (they are both right and share an angle, and thus have the same angle measures for all angles). We therefore have,

x/y = 8/3

8y = 3x

y = 3x/8

Substituting into our area equation,

A = (8 x)(3x/8) = 3x 3x2/8

Taking the derivative,

A' = 3 3x/4 and setting it equal to 0,

3 3x/4 = 0

x = 4

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