MATH SOLVE

2 months ago

Q:
# A rectangular page is to contain 8 square inches of print. The margins at the top and bottom of the page are to be 2 inches wide. The margins on each side are to be 1 inch wide. Find the dimensions of the page that will minimize the amount of paper used. (Let x represent the width of the page and let y represent the height.)

Accepted Solution

A:

Answer: l = 4 in d = 6 in Note : l represent the width and d represent the heightStep-by-step explanation:Lets asume the dimensions of a rectangular page is l * d where l is the wide and d is top-bottom leghtSo total area of the page is A = l*d ⇒ d = A/l ⇒ d = 8/lThen the print area of the page will be l = x + 2 (wide) d = y + 4 (vertical lenght)So area of the page is A(x) = (x + 2 ) * ( y + 4 ) but y = 8/l and l= x +2 ⇒ y = 8 ÷ ( x + 2 )A (x) = ( x + 2) * ( 8 / x + 4 ) ⇒ A(x) = 8 + 4x +16/x + 8 ⇒A(x) = 4x + 16 /xTaken derivative we have:A´(x) = 4 + (-1 *(16)/x² ⇒ A´(x) = 4 - 16/x²A ´(x) = 0 means 4 - 16 /x² = 0 ⇒ 4 x² - 16 = 0 x² = 4 x = 2Therefore y = 8 ÷ ( x +2) and y = 2And the dimensions of the page is l = 4 in d = 6 in