The top and bottom margins of a poster are 8 cm
and the side margins are each 6 cm. If the area of printed material
on the poster is fixed at 380 square centimeters, find the
dimensions of the poster with the smallest area.
find the width and height
width=16.9981
length=22.7084
To find the dimensions of the poster with the smallest area, we need to consider that the area of the printed material is fixed at 380 square centimeters. Let's assume the width of the poster is x and the height is y (both in centimeters).
Given:
Top and bottom margins = 8 cm
Side margins = 6 cm
Area of printed material = 380 square centimeters
To find the dimensions with the smallest area, we need to minimize the total area of the poster, which is the sum of the printed material area and the margins.
Total area of the poster:
Total area = (x + 2*6) * (y + 2*8)
= (x + 12) * (y + 16)
= xy + 16x + 12y + 192
But we know that the area of the printed material is fixed at 380 square centimeters. So,
xy = 380
Now we can express the total area of the poster in terms of a single variable and find the minimum.
Total area = 380 + 16x + 12y + 192
= 572 + 16x + 12y
To find the minimum, we need to differentiate the total area with respect to either x or y and set it to zero. Let's differentiate with respect to x:
d(Total area)/dx = 16 + 0 + 0 [The derivative of a constant is zero]
= 16
Setting this derivative equal to zero to find critical points:
16 = 0
Since the derivative is a constant and can never be zero, we conclude that there are no critical points with respect to x.
Therefore, we can conclude that the width of the poster (x) does not affect its minimum area.
Now, let's differentiate the total area with respect to y:
d(Total area)/dy = 0 + 12 + 0 [The derivative of a constant is zero]
= 12
Setting this derivative equal to zero to find critical points:
12 = 0
Again, since the derivative is a constant and can never be zero, we conclude that there are no critical points with respect to y.
Therefore, we can also conclude that the height of the poster (y) does not affect its minimum area.
Hence, the dimensions of the poster with the smallest area will be the same as the dimensions of the printed material area, which is 380 square centimeters.