profit is the difference between revenue and cost. the revenue, in dollars, of a company that manufactures…

profit is the difference between revenue and cost. the revenue, in dollars, of a company that manufactures televisions can be modeled by the polynomial $3x^2 + 180x$. the cost, in dollars, of producing the televisions can be modeled by $3x^2 - 160x + 300$. the variable $x$ is the number of televisions sold. if 150 televisions are sold, what is the profit? $2,700 $6,000 $50,700 $51,300

profit is the difference between revenue and cost. the revenue, in dollars, of a company that manufactures televisions can be modeled by the polynomial $3x^2 + 180x$. the cost, in dollars, of producing the televisions can be modeled by $3x^2 - 160x + 300$. the variable $x$ is the number of televisions sold. if 150 televisions are sold, what is the profit? $2,700 $6,000 $50,700 $51,300

Answer

Explanation:

Step1: Recall Profit Formula

Profit ( P = \text{Revenue} - \text{Cost} ). Let ( R = 3x^2 + 180x ) (revenue) and ( C = 3x^2 - 160x + 300 ) (cost). So ( P = R - C ).

Step2: Substitute R and C

Substitute ( R ) and ( C ) into the profit formula:
( P=(3x^2 + 180x)-(3x^2 - 160x + 300) ).
Simplify by distributing the negative sign:
( P = 3x^2 + 180x - 3x^2 + 160x - 300 ).
Combine like terms: ( 3x^2 - 3x^2 = 0 ), ( 180x + 160x = 340x ), so ( P = 340x - 300 ).

Step3: Plug in ( x = 150 )

Substitute ( x = 150 ) into ( P = 340x - 300 ):
( P = 340(150) - 300 ).
Calculate ( 340 \times 150 = 51000 ), then ( 51000 - 300 = 50700 )? Wait, no—wait, let's re - check the simplification. Wait, original revenue is ( 3x^2 + 180x ), cost is ( 3x^2 - 160x + 300 ). So ( R - C=(3x^2 + 180x)-(3x^2 - 160x + 300)=3x^2 + 180x - 3x^2 + 160x - 300 = 340x - 300 ). Wait, but when ( x = 150 ), ( 340\times150 = 51000 ), ( 51000 - 300 = 50700 )? But wait, the options have $50,700 as an option. Wait, but let's check again. Wait, maybe I made a mistake in the profit formula? Wait, profit is revenue minus cost. Let's re - compute ( R - C ):

( R = 3x^2+180x ), ( C = 3x^2 - 160x + 300 )

( R - C=(3x^2+180x)-(3x^2 - 160x + 300)=3x^2+180x - 3x^2 + 160x - 300=(3x^2 - 3x^2)+(180x + 160x)-300 = 340x-300 )

Now plug ( x = 150 ):

( 340\times150=51000 ), ( 51000 - 300 = 50700 ). Wait, but the options have $50,700 as one of the options (the third option). Wait, but let's check the arithmetic again. ( 340\times150 ): ( 300\times150 = 45000 ), ( 40\times150 = 6000 ), so ( 45000+6000 = 51000 ). Then ( 51000 - 300 = 50700 ). So the profit is $50,700.

Wait, but let's check the options. The options are $2,700, $6,000, $50,700, $51,300. So the correct answer should be $50,700.

Wait, but wait, maybe I messed up the sign when distributing. Let's do it again:

( (3x^2 + 180x)-(3x^2 - 160x + 300)=3x^2 + 180x - 3x^2 + 160x - 300 ). The ( 3x^2 ) terms cancel. ( 180x+160x = 340x ). Then ( - 300 ). So ( P = 340x - 300 ). For ( x = 150 ), ( 340\times150=51000 ), ( 51000 - 300 = 50700 ). So the profit is $50,700.

Answer:

($50,700) (corresponding to the option with ($50,700))