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Jessica 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 5 cookies from their bag. The product of the number of cookies left in each bag is no more than 80. How many more cookies will Jessica have than Martha?

If [tex]\( x \)[/tex] 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 [tex]\( x^2 \leq \square \)[/tex].

[tex]\( x + 224 \geq \square \)[/tex].

Jessica has at least [tex]\(\square\)[/tex] cookies more than Martha.

Sagot :

GinnyAnswer
Let's analyze the problem step by step based on the provided information:

1. Let [tex]\( x \)[/tex] be the number of cookies Jessica started with.
2. Since the total number of cookies is 30, Martha started with [tex]\( 30 - x \)[/tex] cookies.

After eating 5 cookies each, the number of cookies left for Jessica is [tex]\( x - 5 \)[/tex], and for Martha is [tex]\( 30 - x - 5 = 25 - x \)[/tex].

The problem states that the product of the number of cookies left must be no more than 80:

[tex]\[ (x - 5) \times (25 - x) \leq 80 \][/tex]

Let's expand and simplify this inequality:

[tex]\[ (x - 5)(25 - x) \leq 80 \][/tex]

[tex]\[ 25x - x^2 - 125 + 5x \leq 80 \][/tex]

[tex]\[ -x^2 + 30x - 125 \leq 80 \][/tex]

Subtract 80 from both sides to get a standard quadratic inequality:

[tex]\[ -x^2 + 30x - 205 \leq 0 \][/tex]

Multiply by -1 to make the inequality more familiar:

[tex]\[ x^2 - 30x + 205 \geq 0 \][/tex]

Therefore, the inequality that describes the relationship between the number of cookies is:

[tex]\[ x^2 - 30x + 205 \geq 0 \][/tex]

After eating 5 cookies each, we need to determine how many more cookies Jessica has compared to Martha. This is the difference:

Given the results:

Discriminant: [tex]\( 80 \)[/tex]
Square root of the discriminant: [tex]\( 8.94427190999916 \)[/tex]
Root 1: [tex]\( 19.47213595499958 \)[/tex]
Root 2: [tex]\( 10.52786404500042 \)[/tex]

Take [tex]\( x = 19.47213595499958 \)[/tex]:

[tex]\[ y = 30 - x = 30 - 19.47213595499958 = 10.52786404500042 \][/tex]

Jessica's remaining cookies:

[tex]\[ x - 5 = 19.47213595499958 - 5 = 14.47213595499958 \][/tex]

Martha's remaining cookies:

[tex]\[ y - 5 = 10.52786404500042 - 5 = 5.52786404500042 \][/tex]

The excess cookies Jessica has compared to Martha after eating 5 cookies each is:

[tex]\[ 14.47213595499958 - 5.52786404500042 = 8.94427190999916 \][/tex]

So, Jessica has at least [tex]\( 8.94427190999916 \)[/tex] cookies more than Martha.

Thus, the completed statements are:

The inequality that describes the relationship between the number of cookies each one of them has is [tex]\( x^2 - 30x + 205 \geq 0 \)[/tex].
Jessica has at least [tex]\( 8.94427190999916 \)[/tex] cookies more than Martha.
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