Problem 53 - Algebra and Trigonometry

Question

A cow is tethered to one corner of a square barn, 10 feet by 10 feet, with a rope 100 feet long. What is the maximum grazing area for the cow?

Step-by-Step Solution

Answer

The maximum grazing area for the cow is approximately 7750 square feet.

Step 1: Understand the Problem

A cow is tied to one corner of a square barn measuring 10 feet by 10 feet with a rope that is 100 feet long. We need to find the maximum area the cow can graze.

Step 2: Visualize the Situation

Imagine a square with a side of 10 feet and a circle with a radius of 100 feet from one corner of the square. The cow can graze in a quarter circle initially, but not inside the barn.

Step 3: Calculate the Full Circle Area

The area of a circle with radius 100 feet can be calculated using the formula: \[ A = \pi r^2 \] Here, \[ r = 100 \] feet Therefore, \[ A = \pi (100)^2 = 10000\pi \] square feet.

Step 4: Calculate the Quarter Circle Area

Since the cow starts grazing from a corner: \[ A_{quarter} = \frac{1}{4} \times 10000\pi = 2500\pi \] square feet.

Step 5: Subtract the Area of the Barn

The area of the barn that the cow cannot graze in is: \[ A_{barn} = 10\times10 = 100 \] square feet We need to subtract this from the quarter circle area, so The maximum grazing area \[ = 2500\pi - 100 \] The approximation would then be \[2500 \cdot 3.14 - 100 \] Hence, \[ 7850 - 100 = 7750 \] square feet approximately.

Key Concepts

geometry

In geometry, understanding shapes and their properties is fundamental. In this exercise, we are dealing with a square and a circle. A square has all sides equal, and the internal angles are right angles (90 degrees).

Our square barn is 10 feet by 10 feet. When a cow is tethered to one of the corners of this square, it can graze in a circular area defined by the radius of the rope. For a circle, the radius is the distance from the center to any point on the circumference.

In our problem, this radius is quite large compared to the side of the barn (100 feet of rope vs. 10-foot sides). Visualizing these shapes will help us understand where the cow can graze and how to calculate that area.

circle area calculation

To find the area that the cow can graze, we need to work with the circle's area first. The formula for the area of a circle is \[ A = \pi r^2 \]

Here, \( r \) is the radius. For our problem, the radius \( r \) is 100 feet. Plugging this into our formula, we get: \[ A = \pi (100)^2 = 10000\pi \] square feet.

Next, consider that the cow can only graze in a quarter of this circle because it is tethered to a corner of the square barn. This means we need to find the area of a quarter circle. To do this, we multiply the full circle area by \( \frac{1}{4} \): \[ A_{quarter} = \frac{1}{4} \times 10000\pi = 2500\pi \] square feet.

subtraction of areas

Now, let's consider what area needs to be subtracted from the grazing area. Since the cow cannot graze inside the barn, we need to subtract this part from the quarter circle area.

The area of the barn can be calculated as follows: \[ A_{barn} = 10 \times10 = 100 \] square feet.

Therefore, to find the actual grazing area, we subtract the barn's area from the quarter circle's area: \[ \text{Maximum Grazing Area} = 2500\pi - 100 \]

If we approximate \( \pi \) as 3.14, we get: \[ \approx 2500 \times 3.14 - 100 \approx 7850 - 100 = 7750 \] square feet.

This careful subtraction helps us understand the actual space available for grazing.