Question

Ms. Ahmed has a rectangular garden where she grows tomatoes. The length of her garden is represented by t + 4 and the width by 2t - 1. Express the area of Ms. Ahmed's garden as a polynomial in standard form. If the area of Ms. Ahmed's garden is 56 ft2, what are the dimensions of the garden?

Answer 1

The dimension is 8ft by 7ft

Explanation:

Given data

Length l =
`(t+4)`

Width w=
`(2t-1)`

the area is expressed as
`A= length * width`

`A= (t+4)*(2t-1)\\A= t(2t-1) +4(2t-1)\\A= 2t^(2)-t+8t-4\\`

collecting like terms we have

`A= 2t^(2) +7t-4`

hence our expression for area is
`A= 2t^(2) +7t-4`

given that the area is
`56ft^(2)` to solve for the sides we need to first solve fot t

equating the expression for area to 56 we can solve for t

`A= 2t^(2) +7t-4= 56`

taking the constant term to the other side and solve we have

`A= 2t^(2) +7t-4-56\\A= 2t^(2) +7t-60\\`

we can substitute two factors for 7t that when multiplied we give -60 and when added we give 7 these factors are 15 and -8

`A= 2t^(2) +7t-60\\\\A= 2t^(2) -8t+15t-60\\2t^(2) -8t+15t-60=0\\\\2t(t-4) +15(t-4)= 0\\2t+15=0, (t-4)= 0\\2t=-15, t= -7.5\\t= 4`

t=4 hence the length is

`(t+4)\\(4+4)= 8ft\\`

hence the width is

`(2t-1)\\(2*4-1)= (8-1)= 7ft`

Answer 2

Area of the garden in standard form:

Area = (2t^2) + 7t - 4

The dimension of the garden is 8ft × 7ft

Explanation:

Length of the rectangular garden, L = t+4

Width of the garden, B = 2t - 1

Area = length * width

Area = (t+4) * ( 2t-1)

Area = (2t^2) + 7t - 4..….......(1)

If the Area = 56 ft^2

Substitute Area into (1)

56 = (2t^2) + 7t - 4

(2t^2) + 7t - 60 = 0

(2t^2) - 8t + 15t - 60 = 0

2t(t - 4) + 15(t - 4) = 0

(2t+15) (t-4) = 0

Let 2t + 15 =0

t = -7.5

Let t - 4 = 0

t = 4

* When t = -7.5

Length, L = -7.5 + 4 = -3.5 ft

Width, B = 2(-7.5) -1 = -16 ft

Since length and breadth cannot be negative, t = -7.5 is not possible

** When t = 4

Length, L = 4+4 = 8 ft

Width, B = 2(4) - 1 = 7 ft

Therefore, the dimension of the garden is 8ft × 7ft