Question

Essica and Martha each have a bag of cookies with unequal quantities. They have 30 cookies total between the two of them. Each of them ate 6 cookies from their bag. The product of the number of cookies left in each bag is not more than 80. How many more cookies will Jessica have Martha? If x represents the number of cookies Jessica started with, complete the statements below. The inequality that describes the relationship between the number of cookies each one of them has is x2 - x + 224 ≥ 0. Jessica has at least cookies more than Martha.

Answer 1

I just took the test. For PLATO the first blank is 30 and the second blank is 2.

Answer 2

Part 1) The inequality that describes the relationship between the number of cookies each one of them has is
`x^(2) -30x+224\geq 0`

Part 2) Jessica has at least 2 cookies more than Martha.

Step-by-step explanation:

Let Jessica has x cookies.

Let Martha has y cookies.

Total cookies they have = 30

Equation forms:

`x+y=30` or
`y = 30-x` .....(1)

Each of them ate 6 cookies from their bag.

Cookies left with Jessica =
`x-6`

Cookies left with Martha =
`y-6`

The product of the number of cookies left in each bag is not more than 80.

`(x-6)(y-6) \leq 80` ....(2)

Substituting
`y = 30-x` in equation (2)

`(x-6)(30-x-6) \leq 80`

=>
`(x-6)(24-x) \leq 80`

=>
`24x-x^(2)-144+6x \leq 80`

=>
`-x^(2)+30x-144 \leq 80`

=>
`-x^(2)+30x-224 \leq 0`

Multiplying both sides by -1.

`x^(2) -30x+224 \geq 0`

Solving this quadratic equation, we get

`x\leq 14` or
`x\geq 16`

We will take x = 16 (bigger value as Jessica has more cookies)

And y =
`30-16=14`

y = 14

So, Jessica has 16 cookies.

Martha has 14 cookies.

Cookies left in the bag :

Jessica :
`16-6=10` cookies

Martha :
`14-6=8` cookies

Therefore, Jessica has at least 2 cookies more than Martha.