A construction company has adjoined a 3600 ft^2 rectangular enclosure to its office building. Three sides of the enclosure are fenced in. The side of the building adjacent to the enclosure is 360 ft long and a portion of this side is used as the fourth side of the enclosure. Let x and y be the dimensions of the enclosure, where x is measured parallel to the building, and let L
be the length of fencing required for those dimensions.
(a) Find a formula for L
in terms of x and y
.
L(x,y)=
(b) Find a formula that expresses L
as a function of x
alone.
L(x)=
(c) What is the domain of the function in part (b)? Express as an interval.
I will set it up for you, then you can dig out the parts that fit into
each question
As you suggest, let the length of the enclosure by x
and its width be y
So we need (x + 2y) ft of fencing, since we only need 1 length.
We are told that xy = 3600 , or y = 3600/x
So L = x + 2y
= x + 2(3600/x) = x + 7200/x
Since this is Calculus, I will assume that there is more to this question
such as we are given the cost of the fencing.
A typical follow-up would be to find the dimensions of the enclosure
for a minimum cost of the fencing.
(a) The formula for L in terms of x and y can be found by adding up the lengths of all four sides of the enclosure. The three sides that are fenced in have a combined length of 3600 ft^2. The fourth side, which is one of the sides of the building adjacent to the enclosure, has a length of x ft. Therefore, the formula for L is:
L(x, y) = 3y + x
(b) To express L as a function of x alone, we can eliminate y from the formula. Since y is not given, we can express it in terms of x using the area of the enclosure. The area of the enclosure is given as 3600 ft^2, which can be expressed as:
xy = 3600
Solving for y, we get:
y = 3600/x
Substituting this back into the formula for L:
L(x) = 3(3600/x) + x
(c) The domain of the function L(x) depends on the physical constraints of the problem. In this case, since we're dealing with dimensions, both x and y should be positive values. Furthermore, considering practical constraints, we can assume that the enclosure cannot have a length greater than the length of the side of the building it's attached to (360 ft). Therefore, the domain of the function L(x) is the interval (0, 360].
(a) To find a formula for L in terms of x and y, we need to consider the perimeter of the enclosure.
The perimeter of the enclosure consists of three sides (y + x + y) and the fourth side, which is the length of the building adjacent to the enclosure (360 ft).
Therefore, the formula for L in terms of x and y is:
L(x, y) = x + 2y + 360
(b) To express L as a function of x alone, we can substitute x = 360 - 2y into the formula obtained in part (a):
L(x) = (360 - 2y) + 2y + 360
Simplifying,
L(x) = 720
(c) The domain of the function L(x) is the possible values of x. Since x represents the dimension measured parallel to the building, it must be greater than 0. Therefore, the domain of L(x) is the interval (0, ∞).
To find a formula for L in terms of x and y, we need to determine the perimeter of the rectangular enclosure.
(a) The perimeter of a rectangular enclosure is calculated by adding up the lengths of all its sides. In this case, three sides of the enclosure are fenced in, and the fourth side is formed by the adjacent side of the building, which is 360 ft long.
Since the three fenced sides form a rectangle, the sum of their lengths is 2 times the sum of x and y (2(x + y)).
Adding the length of the building side to the equation, we have L = 2(x + y) + 360.
(b) To express L as a function of x alone, we need to eliminate y from the equation. Since L = 2(x + y) + 360, we can rearrange the equation to solve for y:
2(x + y) = L - 360
x + y = (L - 360)/2
y = (L - 360)/2 - x
Substituting this expression back into the equation for L, we get:
L = 2(x + (L - 360)/2 - x) + 360
L = 2(L - 360)/2 + 360
L = L - 360 + 360
L = L
The formula for L as a function of x alone simplifies to L(x) = L.
(c) The domain of the function L(x) is the set of all possible values for x. Since there are no restrictions or constraints given in the problem statement, the domain is the set of real numbers, which can be expressed as the interval (-∞, ∞).