A billboard is to be made with 100m^2 of printed area and with margins of 2m at the top and bottom and 4m on each side. Find the outside dimensions of the billboard if its total area is to be minimum.
Oh, designing billboards, are we? Well, let's calculate the outside dimensions of this thing to minimize its total area.
Considering that the margins take up 2m at the top, 2m at the bottom, 4m on the left side, and 4m on the right side, we need to subtract those dimensions from the total area.
Total area = printed area + margins area
100m^2 = printed area + (2m + 2m) * (4m + 4m)
Simplifying this equation, we get:
100m^2 = printed area + 4m * 8m
To minimize the total area, we can differentiate it with respect to the printed area and set it to zero:
d(total area)/d(printed area) = 1 + 0
Therefore, the minimum outside dimensions of the billboard would be:
Width = 8m + 8m = 16m
Height = 2m + 100m^2/(8m) = 20m
So, the outside dimensions of the billboard would be 16m x 20m. Now you can go ahead and create the most memorable and hilarious billboard ever!
To find the minimum outside dimensions of the billboard, we need to maximize the printed area.
Let's assume the outside dimensions of the billboard are length (L) and width (W). Since there are margins on all sides, the printed area can be calculated as:
Printed Area = (L - 2 × 4) × (W - 2 × 2)
= (L - 8) × (W - 4)
Now, the total area of the billboard is the sum of the printed area and the margins:
Total Area = L × W = Printed Area + (2 × 4) × (W - 2 × 2) + (L - 2 × 4) × 2
= Printed Area + 4(W - 4) + 2(L - 8)
= Printed Area + 4W - 16 + 2L - 16
= Printed Area + 4W + 2L - 32
Since we want to minimize the total area, we need to maximize the printed area. Hence, we need to find the maximum value of (L - 8) × (W - 4) given the constraint that the printed area should be 100m^2.
To find the maximum value, we can take the partial derivatives of the printed area with respect to L and W and set them equal to zero.
∂(Printed Area)/∂L = 0
∂(Printed Area)/∂W = 0
Using the chain rule, we have:
∂(Printed Area)/∂L = (W - 4) - 2(L - 8) = 0 (1)
∂(Printed Area)/∂W = (L - 8) - 2(W - 4) = 0 (2)
Simplifying the equations, we get:
W - 4 - 2L + 16 = 0 (From equation 1)
L - 8 - 2W + 8 = 0 (From equation 2)
Simplifying further, we have:
W - 2L + 12 = 0 (3)
-2W + L = 0 (4)
Solving equations 3 and 4 simultaneously, we find:
W = 12 and L = 24
Therefore, the outside dimensions of the billboard to achieve the minimum total area is 24m x 12m.
To find the outside dimensions of the billboard that will minimize its total area, we can start by representing the dimensions of the printed area.
Let's assume the length of the printed area is x meters. This means the width of the printed area should be (100 / x) meters, as the printed area has an area of 100 m^2.
Now, to determine the outside dimensions of the billboard, we need to add the margin sizes to the length and width of the printed area. The length of the billboard will be x + 2 + 2 = x + 4 meters, and the width of the billboard will be (100 / x) + 4 + 4 = (100 / x) + 8 meters.
To find the total area of the billboard, we multiply its length by its width:
Total area = (x + 4) * ((100 / x) + 8)
To find the dimensions that will minimize the total area, we can take the derivative of the total area function with respect to x and set it equal to zero:
d(Total area) / dx = 0
Next, we differentiate the total area function and solve for x:
d(Total area) / dx = (x + 4) * (-100 / x^2) + (100 / x) + 8 = 0
Simplifying the equation:
-100(x + 4) + 100(x^2 / x) + 8x^2 = 0
Simplifying further:
-100x - 400 + 100x + 8x^2 = 0
8x^2 - 400 = 0
Solving for x, we find:
x^2 = 50
x = √50
Now that we have the value of x, we can substitute it back into the dimensions of the billboard to find the outside dimensions:
Length of the billboard = x + 4 = √50 + 4
Width of the billboard = (100 / x) + 8 = (100 / √50) + 8
So, the outside dimensions of the billboard that will minimize its total area are approximately:
Length ≈ √50 + 4 meters
Width ≈ (100 / √50) + 8 meters
Length of billboard = x
Width of billboard = (100/x)
Length including margin = x + 4 + 4 = (x+8)
Width including margin = (100/x) + 2 + 2 = (100/x + 4)
The total area is given by:
(x+8)*(100/x + 4)
= 100 + (800/x) + 4x + 32
= 4x + (800/x) + 132
This is the function to be minimized. The derivative of the function is:
f'(x) = 4 - (800/x^2)
Equating this to zero,
4 = (800/x^2)
x^2 = 200
x = √200
= 10√2
This is the required value of 'x' at the minima (the negative value is irrelevant for this problem)
Put the value of x to get the required dimensions.