Question

the top and bottom margins of a poster are each 12 cm and the side margins are each 8 cm. if the area of printed material on the poster is fixed at 1,536 cm2, find the dimensions (in cm) of the poster with the smallest area. 128128

Answer 1

To find the dimensions of the poster with the smallest area when the area of the printed material is fixed at 1536 cm², one must use optimization techniques involving calculus. After adding the margins to the printed area dimensions, one can derive a function for the total area and find its minimum value by setting its derivative to zero and solving for one dimension.

Step-by-step explanation:

The question asks for the dimensions of the entire poster when the area of the printed material is 1536 cm2 and the margins are given. First, let's represent the width and height of the printed area as x and y respectively. The overall width and height of the poster would then be x + 2×8 cm and y + 2×12 cm, since we have to add twice the side and top/bottom margins to the corresponding dimensions.

To find the smallest area of the poster, we can use the fixed area of the printed material to express one variable in terms of the other (i.e., y = 1536 cm2 / x). This allows us to express the area of the entire poster, including the margins, as a function of a single variable:

A(x) = (x + 16) × (1536/x + 24).

We now have a function that can be optimized using calculus or by completing the square. To find the minimum area, we can differentiate this function with respect to x, set the derivative to zero, and solve for x. This will give us the dimension x for which the poster's area is minimized. The corresponding dimension y can then be found using the relation y = 1536 cm2 / x. Substituting these dimensions back into the equations for the overall dimensions of the poster gives us the dimensions of the poster with the smallest area.

Answer 2

To minimize the poster's area with fixed printed material at 1536 cm², the printed area should be a square, leading to a calculation of the square root of 1536 for both dimensions, and adding the respective margins to obtain the total poster dimensions.

Step-by-step explanation:

To find the dimensions of the poster with the smallest area that fits the given area of printed material, we first denote the width and height of the printed area as x and y respectively (in centimeters). Given the area of the printed material is 1536 cm², we set up the equation x×y = 1536.

Considering the margins, the total width of the poster would be x + 2×8 (for both side margins), and the height would be y + 2×12 (for the top and bottom margins). To minimize the area of the poster, we use the method of Lagrange multipliers or optimize the area function A(x, y) = (x+16)(y+24) with the constraint x×y = 1536.

Through optimization, we find that the dimensions of the printed material that give the smallest poster area when accounting for the margins are when x and y are equal, signifying a square shape for the printed area. So, if the printed area is a square, x = y = √1536 cm. Then, the total dimensions of the poster are (width × height): (√1536+16) cm by (√1536+24) cm.