1. What equation would describe the area of the garden? Write the equation in terms of the width of the garden.
2. What can you say about the equation formulated in item 1??
3. Find the roots of the equation formulated in item 1. What does the roots represent?
4. What is the sum of the roots and how this is related in the perimeter?
5. What is the product of the roots and how it this related to area of the rectangle?
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1. Perimeter is twice the sum of the length and width, can be written in P = 2(l + w). Since we already know the perimeter,
56 = 2 (l + w)
Finding the value of width,
56 = 2 (l + w)
28 = l + w
28 - l = w
Area is the product of length and width, can be written as A = lw. Substituting the value of w in this equation,
A = lw
192 = l(28-l)
192 = - l² + 28l
0 = l² - 28l + 192
2. It's a quadratic equation.
3. Finding the roots by factoring,
l² - 28l + 192 = 0
( x-12 ) ( x-16 ) = 0
x = 12
x = 16
4. The sum of the roots is 56, and they are related to the perimeter since 12 and 16 are the roots or the dimensions of the rectangle.
5. The product of the roots is 192, and they are related to the area since the roots are the dimensions itself.
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