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german
3 years ago
11

Dana has 800 yards of fencing to enclose a rectangular area. Find the dimensions of the rectangle that maximize the enclosed are

a. What is the maximum? area
Mathematics
1 answer:
grin007 [14]3 years ago
4 0

Answer:

Perimeter of rectangle states that total distance around the outside, which can be found by adding together the length of each side.

As per the given statement, Dana has 800 yards of fencing to enclose a rectangular area.

Therefore, perimeter of rectangle = 800 yards.

Perimeter(P) of rectangle is given by :

P = 2(x+y) where x is the length and y is the width of the rectangles respectively.

Then,

2(x+y) = 800

Divide both sides by 2 we get;

x+y =400

or

y = 400 -x            .....[1]

We know, the area(A) of rectangle is given by

A = xy                   .....[2]

Substitute equation [1] into [2] we get;l

A = x(400-x)

using distributive property a\cdot (b+c) = a\cdot b+ a\cdot c

A = 400x -x^2            

or

A =-x^2+400x    ......[3]

To maximize the enclosed area.

For a quadratic function in standard form y =ax^2+bx+c , the axis of symmetry is a vertical line given by;  

x = -\frac{b}{2a}

Then; we have from equation [3]

a = -1 and b = 400

By axis of symmetry;

x = -\frac{b}{2a} = -\frac{400}{2 \cdot -1} =\frac{400}{2} = 200

Substitute in equation [1] to solve for y:

y = 400 - 200 = 200

The dimensions of the rectangle that maximize enclosed area are:

x = 200 units and y = 200 units.

Maximum area = 200 \times 200 = 40,000 square units.

We can say that maximum area of rectangle will always be a square



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<em>Comment on vocabulary</em>

<u>mean</u>: the sum of numbers divided by the number of numbers. 5 numbers with a mean of 4.4 will have a sum of 5×4.4 = 22.

<u>median</u>: the middle number of a collection of an odd number of numbers, when sorted from lowest to highest. It is the 3rd number of a group of 5 numbers. When there are an even number of numbers in the group, the median is the average of the middle two in the sorted list. Here, that means the 3rd number in the list of 5 is '4'. When 8 is added, 4 becomes one of the numbers contributing to the average that will be the new median.

<u>mode</u>: the member of the group that appears the most times. When the mode is unique, it means there are no other numbers repeated as often. Here, we know the 3rd number (of 5) is 4, so there can only be two numbers '3' in the collection. That 3 is the unique mode means no other numbers are repeated at all.

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