A rectangular �eld is to be fenced o� along the bank of a river; no fence is required along the bank
of the river. The material for the fence costs 8 dollars per running foot for the two ends and 16
dollars per running foot for the side parallel to the river. If the area of the �eld is to be 12,000
sq. feet and the cost of the fence is not to exceed 3,520 dollars, what are the restrictions on the
dimensions of the �eld?
I'm not sure how to solve this. help me pleasee. thank you
What's an eld ?
let the side parallel to the rive by y ft
let the other sides be x ft each
so xy = 1200
y = 1200/x
cost of fence = 16y + 8(2x)
= 16(1200/x) + 16x
= 19200/x + 16x
d(cos)/dx = -19200/x^2 + 16
= 0 for a min cost.
16x^2 = 19200
Can you finish it?
To solve this problem, you can use a combination of algebra and logical reasoning. Let's break down the information given step by step:
1. The field is rectangular and has a certain area. Let's say its dimensions are length (L) and width (W). Therefore, the area of the field can be expressed as L * W = 12,000 sq. feet.
2. The cost of the fence is given in terms of running feet. Let's calculate the cost for the two ends of the field. Since there are two ends, the cost would be 8 dollars/running foot * 2 ends = 16 dollars/foot. Since the width of the field has no fence along the riverbank, only the length has the cost of 16 dollars/foot.
3. The cost of the fence for the side parallel to the river is 16 dollars/running foot.
4. The total cost of the fence must not exceed 3,520 dollars.
Now, let's use this information to form an equation and solve for the restrictions on the dimensions of the field:
The cost of the fence for the two ends = 16 dollars/foot * L feet = 16L dollars.
The cost of the fence for the side parallel to the river = 16 dollars/foot * W feet = 16W dollars.
The total cost of the fence can be expressed as:
Total cost = Cost for the two ends + Cost for the side parallel to the river
Total cost = 16L + 16W dollars.
According to the problem, the total cost must not exceed 3,520 dollars, so we can write the inequality:
16L + 16W ≤ 3,520.
We also know that the area of the field is 12,000 sq. feet:
L * W = 12,000.
Now, we have a system of equations with two unknowns (L and W) and can solve it using algebraic methods.
The restrictions on the dimensions of the field are the values of L and W that satisfy both the area equation and the cost inequality.
I recommend solving these equations using substitution or elimination method to find the exact restrictions on the dimensions.