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QuestionA sandwich store has a weekly cost of \575.00and$0.65perroastbeefsandwich.Findprofitfunction575.00 and \$0.65 per roast beef sandwich. Find profit function P(x)$ and max profit.

Studdy Solution

STEP 1

Assumptions1. The fixed weekly cost of the sandwich store is $575.00\$575.00 . The variable cost for making a roast beef sandwich is $0.65\$0.65
3. The weekly cost function, C(x)C(x), is given by C(x)=575.00+0.65xC(x) =575.00 +0.65x
4. The revenue function, R(x)R(x), is given by R(x)=−0.001x+3xR(x) = -0.001x^{} +3x
5. The profit function, (x)(x), is the difference between the revenue and cost functions

STEP 2

First, we need to write the profit function, (x)(x), which is the difference between the revenue function, R(x)R(x), and the cost function, C(x)C(x).
(x)=R(x)−C(x)(x) = R(x) - C(x)

STEP 3

Now, plug in the given functions for R(x)R(x) and C(x)C(x) to write the profit function.
(x)=(−0.001x2+3x)−(575.00+0.65x)(x) = (-0.001x^{2} +3x) - (575.00 +0.65x)

STEP 4

implify the profit function.
(x)=−0.001x2+3x−575.00−0.65x(x) = -0.001x^{2} +3x -575.00 -0.65x

STEP 5

Combine like terms.
(x)=−0.001x2+(3−0.65)x−575.00(x) = -0.001x^{2} + (3 -0.65)x -575.00

STEP 6

implify the profit function.
(x)=−0.001x2+2.35x−575.00(x) = -0.001x^{2} +2.35x -575.00

STEP 7

To maximize the profit, we need to find the vertex of the parabola described by the profit function. The xx-coordinate of the vertex of a parabola given by f(x)=ax2+bx+cf(x) = ax^{2} + bx + c is given by −b/2a-b/2a.
xvertex=−b/2ax_{vertex} = -b/2a

STEP 8

Plug in the values for aa and bb from the profit function.
xvertex=−2.35/(2∗−0.001)x_{vertex} = -2.35 / (2 * -0.001)

STEP 9

Calculate the xx-coordinate of the vertex.
xvertex=−2.35/−.002=1175x_{vertex} = -2.35 / -.002 =1175

STEP 10

Now, plug in the xx-coordinate of the vertex into the profit function to find the maximum weekly profit.
max=(xvertex)=−0.001(1175)2+2.35(1175)−575.00_{max} =(x_{vertex}) = -0.001(1175)^{2} +2.35(1175) -575.00

STEP 11

Calculate the maximum weekly profit.
max=−0.001(1175)+.35(1175)−575.00=$1006.25_{max} = -0.001(1175)^{} +.35(1175) -575.00 = \$1006.25The maximum weekly profit is \$1006.25 when1175 sandwiches are sold.

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