The total cost (in dollars) for a company to manufacture and sell
x items per week is C=50x+40 whereas the revenue brought in by selling all
x items is R=68x−0.3x^2. How many items must be sold to obtain a weekly profit of $200?
Profit = Revenue - Cost
So, you want to find x such that
(68x-0.3x^2)-(50x+40) = 200
or, in a more standard form,
0.3x^2-18x+240 = 0
or
x^2-60x+800 = 0
(x-40)(x-20) = 0
...
The revenue for a company selling x products is modeled by the polynomial expression 5x - 0.4x2 dollars. The operating cost for the company is modeled by the polynomial expression 0.6x + 500 dollars.
Show your understanding by writing an expression models the profit in dollars of the company as a function of the number of products sold?
To find the number of items that must be sold to obtain a weekly profit of $200, we need to set up an equation.
The profit is calculated by subtracting the cost from the revenue:
Profit = Revenue - Cost
Given that the cost (C) is 50x + 40 and the revenue (R) is 68x - 0.3x^2, we can write the equation as:
Profit = 68x - 0.3x^2 - (50x + 40)
We want the profit to be $200, so we can set up the equation:
200 = 68x - 0.3x^2 - 50x - 40
Simplifying the equation:
0 = -0.3x^2 + 18x - 240
Now, let's solve the quadratic equation to find the value of x:
0 = -0.3x^2 + 18x - 240
To solve this quadratic equation, we can either factor it or use the quadratic formula. Since factoring is not possible, we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In our equation, a = -0.3, b = 18, and c = -240.
x = (-(18) ± √((18)^2 - 4(-0.3)(-240))) / 2(-0.3)
Simplifying further:
x = (-18 ± √(324 - 288)) / (-0.6)
x = (-18 ± √36) / (-0.6)
Now, let's solve for x using both the plus and minus signs:
x1 = (-18 + √36) / (-0.6)
x2 = (-18 - √36) / (-0.6)
x1 = (-18 + 6) / (-0.6) = 24 / (-0.6) = -40
x2 = (-18 - 6) / (-0.6) = -24 / (-0.6) = 40
The negative value does not make sense in this context since it represents a negative number of items. Therefore, we discard x1 = -40.
The number of items that must be sold to obtain a weekly profit of $200 is 40.
To find the number of items that must be sold to obtain a weekly profit of $200, we first need to set up an equation that represents the profit.
The profit is equal to the revenue minus the cost. So we have:
Profit = Revenue - Cost
Since the revenue is given by the equation R = 68x - 0.3x^2, and the cost is given by the equation C = 50x + 40, we can substitute these into the profit equation:
Profit = (68x - 0.3x^2) - (50x + 40)
We want the profit to be $200, so we can set up the equation:
200 = (68x - 0.3x^2) - (50x + 40)
Now we can simplify and solve the equation to find the value of x.
Rearrange the equation:
200 = 68x - 0.3x^2 - 50x - 40
Combine like terms:
200 = 18x - 0.3x^2 - 40
Rearrange the equation to get the quadratic term on the left side:
0.3x^2 - 18x + 240 = 0
Now we have a quadratic equation in the form ax^2 + bx + c = 0, where a = 0.3, b = -18, and c = 240. We can solve this equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values into the formula:
x = (-(-18) ± √((-18)^2 - 4(0.3)(240))) / (2(0.3))
Simplifying:
x = (18 ± √(324 - 288)) / 0.6
x = (18 ± √36) / 0.6
Now we can calculate the two possible values for x:
x1 = (18 + √36) / 0.6
x2 = (18 - √36) / 0.6
Calculate x1:
x1 = (18 + 6) / 0.6
x1 = 24 / 0.6
x1 = 40
Calculate x2:
x2 = (18 - 6) / 0.6
x2 = 12 / 0.6
x2 = 20
Therefore, to obtain a weekly profit of $200, the company must sell either 40 items or 20 items per week.