A poster is to contain 300 (cm square) of printed matter with margins 10cm at the top and bottom and 5cm at each side. Find the overall dimensions if the total area of the poster is minimum.
If the height is y and the width is x, then
a = (x+10)(300/x + 20)
now find da/dx = 0 for minimum area.
I want this solution
To find the overall dimensions of the poster that minimize the total area, we need to consider the dimensions of the printed matter and the margins.
Let's assume the width of the printed matter is x (cm). Since there are 5 cm margins on each side, the total width of the poster will be x + 2(5) = x + 10 cm.
Similarly, let's assume the height of the printed matter is y (cm). With 10 cm margins at the top and bottom, the total height of the poster will be y + 2(10) = y + 20 cm.
We know that the area of the printed matter is 300 cm^2. Therefore, the area of the poster is:
Total area = (y + 20) * (x + 10)
To find the overall dimensions that minimize the total area, we can differentiate the equation with respect to either x or y, set it to zero, and solve for x or y.
Let's differentiate with respect to x:
d(Total area)/dx = (y + 20)
Setting this equation equal to zero, we get y + 20 = 0, which implies y = -20. However, since y represents the height, it cannot be negative. Thus, this solution is not valid.
Now, let's differentiate with respect to y:
d(Total area)/dy = (x + 10)
Setting this equation equal to zero, we get x + 10 = 0, which implies x = -10. Similarly, since x represents the width, it cannot be negative. Thus, this solution is also not valid.
Therefore, there are no critical points for which the total area is minimized.
However, to find a valid solution that is practical, we can choose values for x and y that satisfy the given conditions (300 cm^2 area and 10 cm top and bottom margins, and 5 cm side margins). For example, let's choose x = 10 cm and y = 30 cm:
The width of the poster is x + 2(5) = 10 + 2(5) = 20 cm.
The height of the poster is y + 2(10) = 30 + 2(10) = 50 cm.
Thus, the overall dimensions of the poster are 20 cm by 50 cm.
To find the overall dimensions of the poster that minimize the total area, we need to consider the printed matter area and the margins separately.
Let's denote the width of the printed matter as w and the height of the printed matter as h. Therefore, the printed matter area is given by:
Printed matter area = w * h
We can find the overall width of the poster by considering the printed matter width and the margins on both sides:
Overall width = printed matter width + margin on the left + margin on the right
Similarly, we can find the overall height of the poster by considering the printed matter height and the margins at the top and bottom:
Overall height = printed matter height + margin at the top + margin at the bottom
Since we are given that the margins at the top and bottom are 10cm each, and the margins on the sides are 5cm each, we can express these dimensions as:
Margin at the top and bottom = 10 cm + 10 cm = 20 cm
Margin on the left and right = 5 cm + 5 cm = 10 cm
Now, the total area of the poster is the sum of the printed matter area and the margin area:
Total area = (printed matter area) + (margin area)
Total area = (w * h) + (Overall width * Margin at the top and bottom) + (Overall height * Margin on the left and right)
To minimize the total area, we need to find the values of w and h that will minimize the total area. We can do this by taking partial derivatives of the total area equation with respect to w and h, and setting them equal to zero:
d(Total area) / dw = 0
d(Total area) / dh = 0
Once we solve these equations, we can substitute the values of w and h back into the overall dimensions equations to find the values of the overall width and height that minimize the total area.
Please note that solving the partial derivatives equations can be a complex mathematical process, and it might be easier to use optimization software or graphing techniques to find the values that minimize the total area.