Question

Alternation Is planning a giveaway rmonth. The monthly budget for the station is decided by the expression616x + 660 and the budget is split evenly between x + 6 things, includingthe giveaway. During the giveaway, the prizes will be given to every x + 6 caller. Thestation usually receives 4x2 + 32x + 60 calls during giveaways.Step 1 of 3: Write a rational expression to determine how much of the budget willgo to the giveaway.

Answer 1

Hello there. To solve this question, we'll have to remember some properties about dividing polynomials and finding rational expressions.

First, the monthly budget for the station is decided by the expression

`616x+600`

and the budget is split evenly between

`x+6\text{ }`

things, including the giveaway.

We know that during the giveaway, the prizes will be given to every

`x+6`

caller. The station usually receives

`4x^2+32x+60`

calls during the giveaway.

We want to determine a rational expression that models how much of the budget will go to the giveaway.

For this, we first have to determine how many

`x+6`

callers are in between the

`4x^2+32x+60`

calls the station received.

Taking the ratio between calls and callers, we get:

`(4x^2+32x+60)/(x+6)`

Notice we can rewrite the numerator as follows:

`(4\cdot(x^2+8x+15))/(x+6)=(4\cdot(x+3)(x+5))/(x+6)`

This expression, even though it cannot be simplified, gives us the number of callers of the type we're interested in.

Now, we have to find how much of the budget will go to the giveaway.

For this, we simply divide the budget by the number of callers we found before:

`(616x+600)/(\left((4(x+3)(x+5))/(x+6)\right))=((154x+150)\cdot(x+6))/((x+3)(x+5))=(154x^2+1074x+900)/(x^2+8x+15)`

This is the rational expression that determines how much of the budget will go to the giveaway.