Question

PLS HELP!!!!

A farmer has a rectangular field that is 70 feet by 45 feet as shown below. Instead of keeping the field mowed, he has decided to tether two
cows and three goats to posts in the field. The cows will be on the upper side and the goats on the lower side of the field. Each animal has the
ability to eat the grass growing in a circular area swept out if the animal stretches the ropes taught and walks in a circle. There are portions of
these circles, though, the animals cannot reach due to the fence around the field.
Using the diagram below, how much of the field (area) can the animals not reach (i.e. the areas of the field not covered in circles)?
HINT: You can print this diagram and measure angles using a protractor. The top two circles are the same size as are the bottom three circles.

Answer 1

This is a challenging problem. To solve it, you need to find the area of the rectangle and subtract the area of the five circles segments. A circle segment is the region between a chord and an arc of a circle. To find the area of a circle segment, you can use this formula:

Asegment=r2/2(0-sin0)

where r is the radius of the circle and θ is the central angle in radians.

To use this formula, you need to know the radius and the central angle of each circle segment. The radius is given by half of the length of the rope that tethers each animal. The central angle can be found by using trigonometry or by measuring it with a protractor.

For example, let’s find the area of the circle segment on the top left corner. The radius is r=15 feet, since the rope is 30 feet long. The central angle can be measured as θ=1.57 radians (or 90 degrees). Plugging these values into the formula, we get:

Asegment​=15^/2​(1.57−sin1.57)

Asegment​=225/5​(1.57−1)

Asegment​=64.13 square feet

You can repeat this process for the other four circle segments, using different values of r and θ. Then, you can add up all the areas of the circle segments and subtract them from the area of the rectangle, which is 70×45=3150 square feet.

The final answer will be the area of the field that the animals cannot reach. I hope this helps you solve the problem.

Explanation:

Answer 2

To determine the area of the field that the animals cannot reach, you will need to calculate the area of the circles that each animal can reach and then subtract that from the total area of the field. First, calculate the radius of each animal's circle. To do this, divide the length of the rope by 2, since this is the distance from the post to the outer edge of the circle. Next, calculate the area of each circle using the formula A = πr^2, where A is the area and r is the radius. Add up the areas of all the circles to find the total area covered by the animals. Finally, subtract the total area covered by the animals from the total area of the field (which is given as 70 feet by 45 feet) to find the area of the field that the animals cannot reach. Note that the area of the field that the animals cannot reach will depend on the length of the ropes, which is not given in the problem statement.