Question

A farmer has a quadrilateral parcel of land with a perimeter of 700 feet. Two opposite angles of that parcel of land are right angles, while the remaining two are not. The farmer wants to do organic farming on that parcel of land. The cost of organic farming on that parcel of land is Rs. 400 per square foot.
Consider the following two additional pieces of information:
1. The length of one of the sides of that parcel of land is 110 feet.
2. The distance between the two corner points where the non-perpendicular sides of that parcel of land intersect is 265 feet.
To determine the amount of money the farmer needs to spend to do organic farming on the entire parcel of land, which of the above additional pieces of information are MINIMALLY SUFFICIENT?

• A. Either of I or II, by itself
• B. II only
• C. The amount cannot be determined even with the additional pieces of information.
• D. I only
• E. I and II together only

Hint:

Understanding the geometry of the land, and knowing either a side length or distance between two corner points, can be sufficient to compute the area and the cost.

Answer 1

The Correct Option is A

To determine which additional piece of information is minimally sufficient for calculating the cost of organic farming on the given parcel of land, we analyze the problem with respect to both statements I and II.

Firstly, let's understand the configuration of the parcel of land: 

• The perimeter of the parcel is 700 feet.
• The parcel is a quadrilateral with two opposite right angles.

Assume the quadrilateral is a kite, where one pair of non-adjacent sides equals 110 feet as given in statement I. The remaining two sides would then be equal to ensure the two right angles.

1. Let's analyze Statement I: "The length of one of the sides of that parcel of land is 110 feet."
   • If one side is 110 feet, and the opposite side is also 110 feet due to symmetry and the right angle property, then there are two right-angled triangles (forming a rectangle) with hypotenuse 110 feet.
   • The perimeter is given as 700 feet, so the remaining two sides sum up to \(700 - 2 \times 110 = 480\) feet.
   • As there are two sides that are equal and form a rectangle, each of these sides measures \(480 / 2 = 240\) feet.
2. Let's analyze Statement II: "The distance between the two corner points where the non-perpendicular sides of that parcel of land intersect is 265 feet."
   • This gives us the diagonal in a kite-shaped configuration. Since the perimeter is 700 feet, the other statement is not necessary because once the diagonal is known, we can set up equations using Pythagoras' Theorem to determine the side lengths. From here, the area is calculable.

With either Statement I or II alone:

• The sides can be calculated, and by understanding the geometry (the parcel forms two right triangles), we can find the area.
• The area, \(A\), of the rectangle plus triangles formed by either statement can be used to calculate the cost:
• \(A = \text{base} \times \text{height} = 110 \times 240 = 26400 \text{ square feet}\)
• Cost = \(26400 \times 400 = 10560000\) Rs.

Conclusion: Thus, either statement I or II alone provides sufficient information to calculate the cost. Hence, the correct answer is "Either of I or II, by itself."

Answer 2

Step 1: Evaluate the first piece of information.
Given the length of one side is 110 feet, we can calculate the area of the land by using the length and the given perimeter.
Step 2: Evaluate the second piece of information.
Given the distance between the corner points, we can use trigonometry or geometry to compute the area of the land.
Conclusion: Either piece of information will help in calculating the area and, thus, the cost of organic farming.
Final Answer: \[ \boxed{\text{(A) Either of I or II, by itself}} \]