rectangular sheet metal is 36 inches wide fom into a drain gutter by turning up
ID: 3114772 • Letter: R
Question
rectangular sheet metal is 36 inches wide fom into a drain gutter by turning up the efthe 36 in Pracie parallel sides. Let x represent the length of each of the Give the restrictions on x. Simplity your answers. Type OA. A(x)-(36-ar an intoger or a decimal A that gives the area of a cross-section of a gutter. O C. A(x) x(x-36) D. A(x)=x(36-2x) The value of x for which A will be a maximum is (Simplifty your answer. Type an integer or a decimal.) The maximum area of the cross-section is (Simplify your answer. Type an integer or a decimal.) square inches. values of x for which the area of a cross-section will be less than 48 square inches are between 0 a (Simplity your answers. Type an integer or a decimal. Round to the nearest hundredth if needed.) ID: 3.8.70-GC and 18. or betweenExplanation / Answer
The restrictions on x are 0 < x < 18 ( 0 is obvious and 2x has to be less than 36)
4.If A(x) < 48, we have x(36-2x) < 48 or, 36x-2x2 < 48 or, 2x2-36x+48 > 0 or, x2-18x+24 > 0 or, (x2-18x+81)-57 > 0 or, (x-9)2 –(57)2 > 0 or, (x-9+57)(x-9-57) >0. Thus, either both (x-9+57) and (x-9-57) are less than 0 or, both(x-9+57) and (x-9-57) are greater than 0. If x-9+57 < 0, then x < 9-57 and when x-9-57< 0, then x < 9+57. Hence in this case, we must have x < 9-57. Further, when x-9+57 > 0, then x > 9-57 and when x-9-57> 0, then x > 9+57.Hence, in this case, we must have x > 9+57. Thus, either x < 9-57 or, x > 9+57. On rounding off to 2 decimal places, we have 57 = 7.55 so that either x < 1.45 inches or x > 16.55 inches. Now, The restrictions on x are 0 < x < 18,hence , either x is between 0 and 1.45 inches or x is between 16.55 and 18 inches.
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