Maximizing Cow Grazing: Solving Area and Quadratic Equations with Tethered Cows

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Homework Help Overview

The problem involves determining the grazing area available to a cow tethered by a 50m rope near a barn measuring 60m by 30m. The setup includes various geometric shapes, including circles and rectangles, that need to be analyzed to find the total area the cow can graze.

Discussion Character

  • Exploratory, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • Participants discuss dividing the grazing area into different regions, including semicircles and quarter circles, to calculate the total area. There is uncertainty about how to accurately represent the areas based on the tethering point and barn corners.

Discussion Status

Participants are actively exploring the problem, with some offering hints about dividing the grazing area into sections. There is a focus on understanding how the tethering point affects the shape of the grazing area, particularly at the corners of the barn.

Contextual Notes

Some participants express confusion about the calculations needed for the areas of the circles involved and how to account for the barn's corners in their reasoning.

Paradiselovek
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Area/Quadratic NEED HELP****

The problem is:

A cow is tethered by a piece of rope 50m long. The rope is fastened to a hook located 10 m from the corner of the longest side of a barn measuring 60m by 30 m. Over how much ground can the cow graze.

I add the information above to the diagramp below:

http://www.freeimagehosting.net/uploads/a875067c70.jpg

I try to find the area of the circle then divide it in half (since it half a circle) but I'm not sure so please help
 
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Start by drawing both the circle (R=50m) and the rectangle, with the center of the circle where you've shown the tether spot. Do you see a way that you could divide that up into regions, to get the various areas that make up the grazing area?

Hint -- I think there are at least 3 pieces that you will calculate the area of differently, and then sum them up for the total grazing area...
 


berkeman said:
Start by drawing both the circle (R=50m) and the rectangle, with the center of the circle where you've shown the tether spot. Do you see a way that you could divide that up into regions, to get the various areas that make up the grazing area?

Hint -- I think there are at least 3 pieces that you will calculate the area of differently, and then sum them up for the total grazing area...

Okay, thank you. Are the 3 pieces in the tether spot include 1/4 of a circle (2 of them)with a radius of 10 m, then 1 (1/2) a circle?I'm wondering if that right, if not can you please give me a bit more hint. Thank you so much ^^

http://www.freeimagehosting.net/uploads/2e83ed54ea.jpg
 
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The hook on the upper side of the barn is one pivot point for an semicircle of 50 m.

When the cow passes below the plane (line) of the upper side of the barn, the corner becomes a pivot point for a smaller circle, of a shorter radius.

Then the cow can pass by the bottom right corner, at a still smaller radius.

Think about what happens are each corner.
 


Astronuc said:
The hook on the upper side of the barn is one pivot point for an semicircle of 50 m.

When the cow passes below the plane (line) of the upper side of the barn, the corner becomes a pivot point for a smaller circle, of a shorter radius.

Then the cow can pass by the bottom right corner, at a still smaller radius.

Think about what happens are each corner.

Hi for the solution above, I draw out my circles, but I still don't know how to find the area of the side circle one. Plz help

http://www.freeimagehosting.net/uploads/c17d20dfb6.jpg
 


If one has a radius, then one can find the area.

On the top the cow sweeps through a semi-circle (half-circle). On the side, the cow would sweep through a quarter circle (from horizontal to vertical) before it turns passes the corner.