Quadratic Formula

Question

Arif's backyard is 2626 by 3535 meters. He wants to put a flower garden in the middle of the? backyard, leaving a strip of grass of uniform width around the flower garden. To be? happy, Arif must have 330330 msquared2 of grass. Under these conditions what will the length and width of the garden? be?

Answer 1

First of all, his backyard is much too large, so I will assume that the numbers you entered were either mistyped or mangled in delivery.

 

Let x be the length of the strip of glass around the garden. The garden will then have length 35 - 2x and width 26 - 2x. It's area will be the product of these two distances. Set this equal to 330:

 

(35 - 2x)(26 - 2x) = 330

 

910 - 70x - 52x + 4x² = 330

 

4x² - 122x + 580 = 0

 

2x² - 61x + 290 = 0

 

This must be solved with the quadratic formula:

 

x = (61 ± √[(-61)² - 4(2)(290)])/(2·2) = (61 ± √1401)/4.

 

Notice that there are two solutions which are both positive.

 

x = (61 + √1401)/8 ≈ 24.61 m   and   x = (61 - √1401)/4 ≈ 5.89 m

 

The first solution gives length 35 - 2(24.61) = -14.21 m , which is meaningless. It can be ignored.

The second solution gives length 23.21 m and width 14.21 m.

 

One can verify that the area is 330 m² to the nearest whole number of square meters.