a landscape architect plans to enclose a 3000 square foot rectangular region in a botanical garden. she will use shrubs costing 25$ per foot along 3 sides and fencing costing 15$ along the fourth. write a function that gives the architect's total cost as a function of the length of one of the sides of the garden.
L = length of side = √3000
3(25L) + 15L = ?
To find the total cost as a function of the length of one side of the garden, we need to consider the cost of the shrubs along three sides and the cost of the fencing along the fourth side.
Let's break down the problem step by step:
1. Determine the dimensions of the rectangular region:
- Let's assume the length of the rectangular region is "L" and the width is "W."
- The area of the rectangular region is given as 3000 square feet: L * W = 3000.
2. Calculate the length of the fourth side:
- Since the area of a rectangle is A = Length * Width, we can express the width as W = A / L.
- As we need to enclose the entire area, the length of the fourth side would be W.
3. Calculate the cost of the shrubs:
- The shrubs cost $25 per foot, and the length of three sides that require shrubs is L + L + W = 2L + W.
- Therefore, the cost of the shrubs would be: Cost of shrubs = 25 * (2L + W).
4. Calculate the cost of the fencing:
- The fencing cost is $15 per foot, and the length of the fourth side is W.
- Therefore, the cost of the fencing would be: Cost of fencing = 15 * W.
5. Calculate the total cost:
- The total cost is the sum of the cost of shrubs and the cost of fencing: Total cost = Cost of shrubs + Cost of fencing.
Based on the above steps, we can write a function in mathematical notation:
total_cost(L) = (25 * (2L + (3000 / L))) + (15 * (3000 / L))
This function takes the length of one side "L" as an input and calculates the total cost accordingly.
Note: The function assumes that the width is determined by the given area of 3000 square feet.