A cow is tethered by a piece of rope 50 meters long to the side of a rectangular barn. The rope is fastened to a hook (on the barn) that is located 20 meters from the corner along the longest side of the barn which measures 70 meters by 20 meters. Over how much ground (outside of the barn) can the cow graze? I have no idea what I am supposed to do, all I know is that the circle , when drawn, with the rope being the radius, the area of the circle is 2500pi. I'm not sure where to go from there. Please help, this is an emergency, thank you.
Well, it seems like you're really in a "moo-p" right now! But don't worry, I'm here to help you out of this predicament.
You've got the right idea with visualizing a circle. The rope's length of 50 meters represents the radius of the circle. So, the area of the circle being 2500π square meters is correct.
To find out how much ground the cow can graze, you need to calculate the area of the sector formed by the circle and subtract the area of the right-angled triangle formed inside the barn.
1. Let's start with the circle's area: A = πr^2
Since the rope's length is the radius, we have:
A = π(50)^2 = 2500π
2. Now, let's calculate the area of the right-angled triangle inside the barn:
The barn dimensions are 70 meters by 20 meters, and the rope is tethered 20 meters away from one corner. So, the triangle's base will be 70 - 20 = 50 meters, and its height will be 20 meters.
A = (base * height) / 2
= (50 * 20) / 2
= 500 square meters
3. Finally, subtract the triangle's area from the circle's area to find the grazing ground:
Grazing ground = Circle's area - Triangle's area
= 2500π - 500
= 2500π - 500
= 2000π square meters
So, the cow can graze over an "udderly" delightful area of 2000π square meters outside of the barn! Keep in mind, that's assuming the cow won't get distracted by any particularly delicious blades of grass while grazing. Hope that helps, and happy grazing to your emergency cow!
To determine how much ground outside of the barn the cow can graze, we need to find the area of the circular region formed by the rope.
Given that the area of the circle is 2500π, we can solve for the radius of the circle.
The formula for the area of a circle is A = πr², where A is the area and r is the radius.
By rearranging the formula, we have r = √(A/π).
In this case, A = 2500π, so substituting the values, we get r = √(2500π/π) = √2500 = 50.
This means the radius of the circular region is 50 meters.
To calculate the total area of the circular region, we can use the formula for the area of a circle: A = πr².
Substituting the value of the radius, we have A = π(50)² = 2500π.
Therefore, the cow can graze over 2500π square meters of ground outside of the barn.
To solve this problem, you are correct in realizing that a circle can be drawn with the rope as the radius. The area of this circle will be 2500π square meters.
The rope is tethered 20 meters away from one corner of the barn, which means that the center of the circle is also located 20 meters away from that same corner.
Now, let's consider the rectangular barn. Its dimensions are given as 70 meters by 20 meters. The corner where the rope is tethered to the barn is 20 meters away from both sides that form the dimensions of the barn.
To find the available grazing area, we need to calculate the area outside of the barn but within the circle.
The area of the barn is simply 70 meters multiplied by 20 meters, which gives us 1400 square meters.
The entire area enclosed by the circle is given as 2500π square meters. Therefore, the area outside the barn but within the circle is (2500π - 1400) square meters.
To solve this problem, you just need to subtract the area of the barn from the total area enclosed by the circle.
(2500π - 1400) square meters is the amount of ground the cow can graze outside of the barn.