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A vendor sells pretzels on a city street corner.

He models the monthly costs, in dollars, to operate his business with the function [tex]C(x)[/tex], where [tex]x[/tex] is the number of pretzels he sells:

[tex]\[ C(x) = 0.63x + 800 \][/tex]

Because the vendor sells the pretzels for [tex]\$2.75[/tex] each, he models his monthly revenue from selling [tex]x[/tex] pretzels with the function [tex]R(x)[/tex]:

[tex]\[ R(x) = 2.75x \][/tex]

The function [tex]P(x)[/tex] represents the vendor's monthly profit, in dollars, when selling [tex]x[/tex] pretzels.

[tex]\[ P(x) = R(x) - C(x) \][/tex]

In a month where he sells 600 pretzels, the vendor will make a profit of [tex]P(600)[/tex] dollars. Calculate this profit.

Sagot :

GinnyAnsweridn
To determine the vendor's monthly profit when selling 600 pretzels, let's break down the problem step by step.

1. Calculate the Total Cost (C(x)) for 600 pretzels:
The cost function is given by:
[tex]\[ C(x) = 0.63x + 800 \][/tex]
Substituting [tex]\(x = 600\)[/tex]:
[tex]\[ C(600) = 0.63(600) + 800 = 378 + 800 = 1178 \][/tex]
So, the total cost for 600 pretzels is [tex]$1178. 2. Calculate the Total Revenue (R(x)) for 600 pretzels: The revenue function is given by: \[ R(x) = 2.75x \] Substituting \(x = 600\): \[ R(600) = 2.75(600) = 1650 \] So, the total revenue from selling 600 pretzels is $[/tex]1650.

3. Calculate the Profit (P(x)) for 600 pretzels:
Profit is given by the difference between revenue and cost:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substituting the values for revenue [tex]\(R(600) = 1650\)[/tex] and cost [tex]\(C(600) = 1178\)[/tex]:
[tex]\[ P(600) = 1650 - 1178 = 472 \][/tex]

Therefore, when the vendor sells 600 pretzels in a month, he makes a profit of $472.
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