A rectangular dog run is to be enclosed by a fence and then divided into two smaller rectangular areas by a fence parallel to one of the sides. If 54m of fence is available, find the dimensions of the dog run of greatest possible area. State the total area
Let x be the shorter side.
Longer side = (54-3x)/2 = 27-1.5x
Area, A = x(27-1.5x)=27x-1.5x²
To get maximum area, diff. w.r.t. x and equate to zero.
dA/dx = 27-3x=0
x=9m
longer side = 27-9*1.5=13.5m
The dog run is 9m by 13.5 m total
Total area= 121.5 m²
To find the dimensions of the dog run that will give the greatest possible area, we need to use calculus and optimization techniques.
Let's denote the length of the rectangular dog run as L and the width as W. We're given that the dog run will be divided into two smaller rectangular areas by a fence parallel to one of the sides. Since the fence will run along the width, we can say that the width of each of the smaller rectangular areas is W/2.
Now, let's calculate the total amount of fence required to enclose the dog run:
- Two lengths: 2L
- Three widths (two for dividing the dog run and one for the remaining side): 3W/2
According to the problem, we know that 54m of fence is available. So we can write the equation:
2L + 3W/2 = 54
To find the dimensions of the dog run that maximize the area, we need an equation that relates the area (A) of the dog run to its dimensions.
The area of the dog run can be calculated as the product of its length and width:
A = LW
We can use a technique called "substitution" to express one variable in terms of the other. From our previous equation:
3W/2 = 54 - 2L
Simplifying:
3W = 108 - 4L
W = (108 - 4L)/3
Now, substitute this expression for W in terms of L into the equation for the area:
A = L * [(108 - 4L)/3]
To find the dog run's dimensions that maximize the area, we need to take the derivative of A with respect to L, set it equal to zero, and solve for L:
dA/dL = 0
To solve this equation, we differentiate the equation for A with respect to L:
dA/dL = (108 - 8L)/3
Set it equal to zero:
(108 - 8L)/3 = 0
Simplifying, we find:
108 - 8L = 0
8L = 108
L = 13.5
Now that we have L, we can plug it back into our equation for W to find its value:
W = (108 - 4L)/3 = (108 - 4(13.5))/3 = 14
Therefore, the dimensions for the dog run of greatest possible area are L = 13.5m and W = 14m.
To find the total area, we substitute these values back into the equation for A:
A = (13.5)(14) = 189m²
The total area of the dog run of greatest possible area is 189m².