A rectangle field that is 119 square yards in area is to covered by sod. it is known that the field is 3 yards longer than twice the width. find the dimensions of the field. how do i do this?

Answer 1

It is given that the length is twice the width plus 3 yards; this can be written as a formula:

Length = 2 × Width + 3

To calculate the area of a rectangle the formula is:

Area_rectangle = Length × Width

Substitute for Length in the above formula by the given expression involving Width

→ Area = Length × Width = (2 × Width + 3) × Width = 2 × Width² + 3 × Width

But the area is given as 119 sq yd. If I now write w for Width, this gives:

2w² + 3w = 119 sq yd

→ 2w² +3w - 119 = 0

Which is a quadratic which can be solved either by using the formula or by factorising:

It does factorise: the factor pairs of 119 are 1 × 119 and 7 × 17. One bracket will have (2w ...) and the other will have (w...). As the 119 is negative, there are effectively four pairs of numbers that could be fitted into the brackets: {-1, 119}; {-7, 17}; {7, -17}; {1, 119}. There is one pair which will fit so that when multiplied out gives the quadratic.

A quadratic will always have two solutions; as one of the numbers above is negative and one positive, the two solutions will comprise of one positive and one negative value for w (Width); as a length must be positive, the negative solution cannot work and the positive solution is the required Width, from which the Length can be calculated (as above).