QuestionFind the number of roast beef sandwiches to maximize profit from $P(x)=-0.001 x^{2}+2.45 x-515.00$ . What is the max profit?

Studdy Solution

STEP 1

Assumptions1. The profit function of the store is given by $(x)=-0.001 x^{}+.45 x-515.00$ . $x$ represents the number of roast beef sandwiches made and sold each week3. We are asked to find the number of sandwiches ( $x$ ) that maximizes the profit and the maximum weekly profit

STEP 2

The profit function is a quadratic function of the form $ax^2 + bx + c$ , where $a<0$ . The maximum value of a quadratic function occurs at its vertex.
The x-coordinate of the vertex of a quadratic function $ax^2 + bx + c$ is given by $-b/2a$ .

STEP 3

Now, plug in the given values for $a$ and $b$ to calculate the x-coordinate of the vertex.
$a = -0.001$ and $b =2.45$
The x-coordinate of the vertex is given by $x = -b/2a$ .

STEP 4

Calculate the x-coordinate of the vertex.
$x = -2.45 / (2 \times -0.001)$

STEP 5

implify the expression to find the number of sandwiches that maximizes the profit.
$x = -2.45 / -0.002 =1225$
So, the store should make and sell1225 sandwiches each week to maximize profit.

STEP 6

Now, we need to find the maximum weekly profit. We can do this by substituting the value of $x$ that maximizes profit into the profit function.
$(x) = -0.001 x^{2} +2.45 x -515.00$

STEP 7

Substitute $x =1225$ into the profit function to calculate the maximum weekly profit.
$(1225) = -0.001 \times1225^{2} +2.45 \times1225 -515.00$

STEP 8

Calculate the maximum weekly profit.
$(1225) = -0.001 \times1500625 +2.45 \times1225 -515.00 = \$1495.625$
The maximum weekly profit is \$1495.63 (rounded to two decimal places) when1225 sandwiches are sold.

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