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The chef at the Crescent Café is adding a new sandwich to the menu. Sales data from a similar menu item indicates that if the chef charges x dollars per sandwich, she will sell –20x+490 of the new sandwiches in the first month of their introduction.The cost to make each of the new sandwiches is $3. So, the chef will earn x–3 dollars in profit per sandwich.Which equation can the chef use to predict the price she can charge per sandwich to earn $2,000 in profit that first month?2,000=(x–3)(–20x+490)2,000x=(x–3)(–20x+490)To the nearest dollar, what is the lowest price the chef can charge per sandwich to earn $2,000 in profit that first month?

Question

The chef at the Crescent Café is adding a new sandwich to the menu. Sales data from a similar menu item indicates that if the chef charges x dollars per sandwich, she will sell –20x+490 of the new sandwiches in the first month of their introduction.The cost to make each of the new sandwiches is 3.So,thechefwillearnx3dollarsinprofitpersandwich.Whichequationcanthechefusetopredictthepriceshecanchargepersandwichtoearn3. So, the chef will earn x–3 dollars in profit per sandwich.Which equation can the chef use to predict the price she can charge per sandwich to earn 2,000 in profit that first month?2,000=(x–3)(–20x+490)2,000x=(x–3)(–20x+490)To the nearest dollar, what is the lowest price the chef can charge per sandwich to earn $2,000 in profit that first month?

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Solution

The equation the chef can use to predict the price she can charge per sandwich to earn $2,000 in profit that first month is 2,000=(x–3)(–20x+490).

To find the lowest price the chef can charge per sandwich to earn $2,000 in profit that first month, we need to solve this equation for x.

First, distribute (x-3) to both terms inside the parentheses:

2000 = -20x^2 + 490x - 3*(-20x) + 3*490 2000 = -20x^2 + 490x + 60x - 1470 2000 = -20x^2 + 550x - 1470

Next, add 1470 to both sides of the equation to set it equal to zero:

3470 = -20x^2 + 550x 0 = -20x^2 + 550x - 3470

Now, divide the entire equation by -20 to simplify:

0 = x^2 - 27.5x + 173.5

This is a quadratic equation, and we can solve it using the quadratic formula:

x = [27.5 ± sqrt((27.5)^2 - 41173.5)] / (2*1) x = [27.5 ± sqrt(756.25 - 694)] / 2 x = [27.5 ± sqrt(62.25)] / 2 x = [27.5 ± 7.89] / 2 x = [19.61, 35.39]

Since the price cannot be negative, the lowest price the chef can charge per sandwich to earn 2,000inprofitthatfirstmonthis2,000 in profit that first month is 20 when rounded to the nearest dollar.

This problem has been solved

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