Let's break down the problem step-by-step to determine the profit when 150 televisions are sold.
### Step 1: Identify the formulas for revenue and cost
The revenue [tex]\( R(x) \)[/tex] and cost [tex]\( C(x) \)[/tex] are given by the following polynomials where [tex]\( x \)[/tex] is the number of televisions sold:
- Revenue: [tex]\( R(x) = 3x^2 + 180x \)[/tex]
- Cost: [tex]\( C(x) = 3x^2 - 160x + 300 \)[/tex]
### Step 2: Substitute [tex]\( x = 150 \)[/tex] into the revenue and cost formulas
First, let's calculate the revenue when [tex]\( x = 150 \)[/tex]:
[tex]\[ R(150) = 3(150)^2 + 180(150) \][/tex]
To break this down:
1. Calculate [tex]\( 150^2 \)[/tex]:
[tex]\[ 150^2 = 22500 \][/tex]
2. Multiply by 3:
[tex]\[ 3 \times 22500 = 67500 \][/tex]
3. Multiply 180 by 150:
[tex]\[ 180 \times 150 = 27000 \][/tex]
4. Add these results together to find the revenue:
[tex]\[ R(150) = 67500 + 27000 = 94500 \][/tex]
So, the revenue for selling 150 televisions is [tex]\( \$94,500 \)[/tex].
Next, let's calculate the cost when [tex]\( x = 150 \)[/tex]:
[tex]\[ C(150) = 3(150)^2 - 160(150) + 300 \][/tex]
To break this down:
1. We already know [tex]\( 150^2 = 22500 \)[/tex].
2. Multiply by 3:
[tex]\[ 3 \times 22500 = 67500 \][/tex]
3. Multiply 160 by 150:
[tex]\[ 160 \times 150 = 24000 \][/tex]
4. Subtract this result from the previous step and then add 300:
[tex]\[ 67500 - 24000 + 300 = 43800 \][/tex]
So, the cost for producing 150 televisions is [tex]\( \$43,800 \)[/tex].
### Step 3: Calculate the profit
Now, we need to find the profit [tex]\( P \)[/tex] which is the difference between the revenue and the cost:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substitute [tex]\( x = 150 \)[/tex] into the profit formula:
[tex]\[ P(150) = 94500 - 43800 \][/tex]
Calculate the profit:
[tex]\[ P(150) = 50700 \][/tex]
### Step 4: Conclusion
The profit when 150 televisions are sold is [tex]\( \$50,700 \)[/tex]. Therefore, the correct answer is [tex]\( \$50,700 \)[/tex].
