Let's break down the calculation step-by-step to determine the profit:
1. Revenue Calculation:
- The revenue function is given by the polynomial [tex]\( R(x) = 3x^2 + 180x \)[/tex].
- We need to calculate the revenue when 150 televisions are sold, i.e., [tex]\( x = 150 \)[/tex].
- Substitute [tex]\( x = 150 \)[/tex] into the revenue function:
[tex]\[
R(150) = 3(150)^2 + 180(150)
\][/tex]
- Calculate [tex]\( 3(150)^2 = 3(22500) = 67500 \)[/tex].
- Calculate [tex]\( 180(150) = 27000 \)[/tex].
- Add the two results: [tex]\( 67500 + 27000 = 94500 \)[/tex].
- Therefore, the revenue from selling 150 televisions is [tex]\( \$94500 \)[/tex].
2. Cost Calculation:
- The cost function is given by the polynomial [tex]\( C(x) = 3x^2 - 160x + 300 \)[/tex].
- We need to calculate the cost when 150 televisions are produced, i.e., [tex]\( x = 150 \)[/tex].
- Substitute [tex]\( x = 150 \)[/tex] into the cost function:
[tex]\[
C(150) = 3(150)^2 - 160(150) + 300
\][/tex]
- Calculate [tex]\( 3(150)^2 = 3(22500) = 67500 \)[/tex].
- Calculate [tex]\( -160(150) = -24000 \)[/tex].
- Add the three results: [tex]\( 67500 - 24000 + 300 \)[/tex].
- Simplify: [tex]\( 67500 - 24000 = 43500 \)[/tex].
- Add [tex]\( 300 \)[/tex]: [tex]\( 43500 + 300 = 43800 \)[/tex].
- Therefore, the cost of producing 150 televisions is [tex]\( \$43800 \)[/tex].
3. Profit Calculation:
- Profit is determined by the difference between revenue and cost: [tex]\( P = R(x) - C(x) \)[/tex].
- Substitute the values we calculated:
[tex]\[
P = 94500 - 43800
\][/tex]
- Calculate the difference: [tex]\( 94500 - 43800 = 50700 \)[/tex].
- Therefore, the profit from selling 150 televisions is [tex]\( \$50700 \)[/tex].
Conclusion: The correct answer is [tex]\( \$50700 \)[/tex], so the correct choice is:
[tex]\(\$ 50,700\)[/tex].
