A page is to contain 54 square centimeters of printed material. If the margins are 1 cm at the top and bottom and 0.5 cm at the sides, find the most economical width of the page (in cm).
w = width
54/w = height
The page area is thus
a = (w+1)(54/w + 2)
Assuming you want minimum page area,
da/dw = 0 at w = 3√3
To find the most economical width of the page, we need to maximize the area of the printed material while minimizing the total area of the page.
Let's calculate the total area of the page with given margins. The top and bottom margins are both 1 cm, and the side margins are 0.5 cm each. So the total width of the margins is 2 * 0.5 cm + 2 * 1 cm = 3 cm.
The total area of the page is the area of the printed material plus the area of the margins. Since the printed material needs to occupy 54 square centimeters, the total area of the page should be equal to 54 square centimeters plus the area of the margins.
Therefore, the total area is (54 + area of margins).
The total width of the page would be equal to the width of the printed material plus twice the side margins.
So the total width is (width of printed material + 2 x side margins).
Now, let's express the total area and total width in terms of the width of the printed material (W).
Total area = (54 + area of margins)
= (54 + (2 x 1 cm x W) + (2 x 0.5 cm x W))
= (54 + 1 cm x W + 0.5 cm x W)
Total width = (width of printed material + 2 x side margins)
= (W + 2 x 0.5 cm)
Since we want to find the most economical width, we need to minimize the total area of the page.
To do that, we differentiate the total area equation with respect to W and set it equal to 0.
d(total area)/dW = 0 + 1 cm + 0.5 cm
1 + 0.5 = 0
1.5 = 0
Since we cannot have a distance equal to 0, it means there is no minimum value of total area.
Hence, we conclude that there is no most economical width of the page in this scenario.
To find the most economical width of the page, we need to maximize the area of the printed material while minimizing the total area of the page.
Let's start by calculating the total area of the page, including margins.
The top and bottom margins have a width of 1 cm each, so their total width is 2 cm.
The side margins have a width of 0.5 cm each, so their total width is 1 cm.
Therefore, the total width of the margins is 2 cm + 1 cm = 3 cm.
The total height of the margins is 1 cm + 1 cm = 2 cm.
So the total area of the margins is 3 cm x 2 cm = 6 square cm.
Now, let's calculate the area of the printed material without the margins.
The total area of the page is 54 square cm, and the area of the margins is 6 square cm.
Therefore, the area of the printed material is 54 square cm – 6 square cm = 48 square cm.
Now, let's denote the width of the printed material as x cm.
Since the width of the page includes the margins, the total width of the page will be x cm + 3 cm.
The height of the printed material is the same as the total height of the page, which is 2 cm.
So, the area of the printed material is x cm x 2 cm = 48 square cm.
Now, we can solve for x.
x x 2 = 48
x = 48 / 2
x = 24
Therefore, the most economical width of the page is 24 cm.