To find the profit when 150 televisions are sold, we need to determine both the revenue and the cost, and then subtract the cost from the revenue. Let's go through the steps in detail.
Step 1: Calculate the Revenue
The revenue function given is:
[tex]\[ \text{Revenue} = 3x^2 + 180x \][/tex]
We need to find the revenue when [tex]\( x = 150 \)[/tex]:
[tex]\[ \text{Revenue} = 3(150)^2 + 180(150) \][/tex]
Perform the calculations:
[tex]\[ 3(150)^2 = 3 \times 22500 = 67500 \][/tex]
[tex]\[ 180(150) = 27000 \][/tex]
So, the revenue is:
[tex]\[ \text{Revenue} = 67500 + 27000 = 94500 \][/tex]
Step 2: Calculate the Cost
The cost function given is:
[tex]\[ \text{Cost} = 3x^2 - 160x + 300 \][/tex]
We need to find the cost when [tex]\( x = 150 \)[/tex]:
[tex]\[ \text{Cost} = 3(150)^2 - 160(150) + 300 \][/tex]
Perform the calculations:
[tex]\[ 3(150)^2 = 3 \times 22500 = 67500 \][/tex]
[tex]\[ 160(150) = 24000 \][/tex]
So, the cost is:
[tex]\[ \text{Cost} = 67500 - 24000 + 300 = 43800 \][/tex]
Step 3: Calculate the Profit
Profit is the difference between revenue and cost:
[tex]\[ \text{Profit} = \text{Revenue} - \text{Cost} \][/tex]
[tex]\[ \text{Profit} = 94500 - 43800 = 50700 \][/tex]
Therefore, the profit when 150 televisions are sold is:
[tex]\[ \boxed{50700} \][/tex]
This matches the option [tex]$\$[/tex] 50,700$.
