Math

QuestionProfit function for a bracelet company is P(x)=2(x2000)P(x)=2(x-2000).
(a) Find P(1000)P(1000) and explain its meaning. (b) Solve P(x)=0P(x)=0 and explain its meaning. (c) Determine the yy-intercept of PP and explain its meaning.

Studdy Solution

STEP 1

Assumptions1. The profit function for the company is (x)=(x2000)(x)=(x-2000). xx represents the number of bracelets sold3. (x)(x) represents the profit in dollars

STEP 2

(a) To find (1000)(1000), we need to substitute x=1000x=1000 into the profit function (x)(x).
(1000)=2(10002000)(1000) =2(1000-2000)

STEP 3

Calculate (1000)(1000).
(1000)=2(10002000)=2(1000)=2000(1000) =2(1000-2000) =2(-1000) = -2000

STEP 4

(b) To solve (x)=0(x)=0, we need to set the profit function (x)(x) equal to zero and solve for xx.
0=2(x2000)0 =2(x-2000)

STEP 5

olve for xx.
0=2(x2000)x2000=0x=20000 =2(x-2000) \Rightarrow x-2000 =0 \Rightarrow x =2000

STEP 6

(c) To determine the yy-intercept of the graph of $$, we need to find the value of $(x)$ when $x=0$.
(0)=2(02000)(0) =2(0-2000)

STEP 7

Calculate (0)(0).
(0)=2(02000)=2(2000)=4000(0) =2(0-2000) =2(-2000) = -4000

STEP 8

Interpretation of results(a) (1000)=2000(1000) = -2000 means that if the company sells1000 bracelets, they will make a loss of $2000.
(b) (x)=0(x)=0 when x=2000x=2000 means that the company will break even (i.e., make neither a profit nor a loss) when they sell2000 bracelets.
(c) The yy-intercept of the graph of $$ is $(0) = -4000$. This means that if the company does not sell any bracelets, they will make a loss of $4000. This could be due to fixed costs that the company has to pay regardless of the number of bracelets sold.

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