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A farmer is trying to maximize the grazing area for their animals. They have 1,000 feet of fencing. They want to place the fencing so that they have created the maximum possible square footage for their animals to graze. They want to use four straight sides to create the grazing area. Part One: What's the maximum possible square footage for animals to graze, given that 1,000 feet of fencing is available? (Hint: The area is maximized if it is built in a square) Part Two: The farmer decides to change the plot of land to a location where a river can be used for fencing. The river is a boundary for the animals. They cannot cross it. The river is very long, so the farmer can use as much of the river as they want for the boundary. The area of the grazing area will be a rectangle. Create an equation that equates the perimeter, 1,000 feet, to the length l and the width w. Part Three: Use your perimeter equation and solve for the variable w. In other words, rearrange the equation so that it looks like w = ... Part Four: Since the area of the grazing land is a rectangle, the area, A, can be calculated using the following equation: A = l * w. Plug in your equation for the letter w (from part three) into the equation for the area of the rectangle. You should now only have the variables A and l in your newly constructed equation. Part Five: Note that your equation in part four is quadratic in terms of l. Quadratics are maximized or minimized at their vertex. Find the length required in order to maximize the grazing area. To do this, you will need to find the vertex of the quadratic. You could expand your equation in part four so that it's of this form: A = al^2 + bl + c where "a", "b", and "c" are constants/numbers that result from rearrangement. Then use the equation to find the vertex: l = -b / (2a) Part Six: What should the farmer plan on using for the width of this rectangle? What is the maximum area of this fence?

A farmer is trying to maximize the grazing area for their animals. They have 1,000 feet of fencing. They want to place the fencing so that they have created the maximum possible square footage for their animals to graze. They want to use four straight sides to create the grazing area.

Part One: What's the maximum possible square footage for animals to graze, given that 1,000 feet of fencing is available? (Hint: The area is maximized if it is built in a square)

Part Two: The farmer decides to change the plot of land to a location where a river can be used for fencing. The river is a boundary for the animals. They cannot cross it. The river is very long, so the farmer can use as much of the river as they want for the boundary. The area of the grazing area will be a rectangle. Create an equation that equates the perimeter, 1,000 feet, to the length l and the width w.

Part Three: Use your perimeter equation and solve for the variable w. In other words, rearrange the equation so that it looks like w = ...

Part Four: Since the area of the grazing land is a rectangle, the area, A, can be calculated using the following equation: A = l * w. Plug in your equation for the letter w (from part three) into the equation for the area of the rectangle. You should now only have the variables A and l in your newly constructed equation.

Part Five: Note that your equation in part four is quadratic in terms of l. Quadratics are maximized or minimized at their vertex. Find the length required in order to maximize the grazing area. To do this, you will need to find the vertex of the quadratic. You could expand your equation in part four so that it's of this form: A = al^2 + bl + c where "a", "b", and "c" are constants/numbers that result from rearrangement. Then use the equation to find the vertex: l = -b / (2a)

Part Six: What should the farmer plan on using for the width of this rectangle? What is the maximum area of this fence?

Added by Helen V.

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