Mara carried water bottles to the field to share with her team at halftime. The water bottles weighed a total of [tex]$60 x^2+[/tex] [tex]$48 x+24$[/tex] ounces.
Which factorization could represent the number of water bottles and weight of each water bottle?
[tex]$6\left(10 x^2+8 x+2\right)$[/tex]
[tex]$12\left(5 x^2+4 x+2\right)$[/tex]
[tex]$6 x\left(10 x^2+8 x+2\right)$[/tex]
[tex]$12 x\left(5 x^2+4 x+2\right)$[/tex]
Answer (2)
The correct factorization for the total weight of the water bottles is 12 ( 5 x 2 + 4 x + 2 ) . This matches option 2 provided in the choices. Therefore, that is the correct answer.
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Find the greatest common divisor (GCD) of the coefficients of the given expression.
Factor out the GCD from the expression: 60 x 2 + 48 x + 24 = 12 ( 5 x 2 + 4 x + 2 ) .
Compare the factored expression with the given options.
The correct factorization is 12 ( 5 x 2 + 4 x + 2 ) .
Explanation
Understanding the Problem We are given the total weight of the water bottles as 60 x 2 + 48 x + 24 ounces. We need to find a factorization that could represent the number of water bottles and the weight of each water bottle. This means we need to factor the given expression and see which of the options matches our factored form.
Factoring the Expression First, we find the greatest common divisor (GCD) of the coefficients 60, 48, and 24. The GCD is 12. We can factor out 12 from the expression:
60 x 2 + 48 x + 24 = 12 ( 5 x 2 + 4 x + 2 )
Comparing with Options Now, we compare our factored expression 12 ( 5 x 2 + 4 x + 2 ) with the given options:
6 ( 10 x 2 + 8 x + 2 )
12 ( 5 x 2 + 4 x + 2 )
6 x ( 10 x 2 + 8 x + 2 )
12 x ( 5 x 2 + 4 x + 2 )
We see that the second option, 12 ( 5 x 2 + 4 x + 2 ) , matches our factored expression.
Final Answer Therefore, the factorization that could represent the number of water bottles and the weight of each water bottle is 12 ( 5 x 2 + 4 x + 2 ) .
Examples
Imagine you are organizing a sports event and need to distribute water bottles to the participants. The total weight of the water bottles can be expressed as a quadratic expression, like the one in the problem. Factoring this expression helps you determine the number of water bottles and the weight of each bottle, ensuring you have the right amount for all participants. This is useful for budgeting and logistics, allowing you to plan effectively and avoid shortages or waste. By understanding the relationship between the total weight, the number of bottles, and the weight per bottle, you can optimize your resources and make informed decisions.
