Question

1. a company that manufactures jeans estimates that the profit for selling a particular style is given by the equation ( p = 250x^3 - 1005x^2 - 300 ), for ( 0 < x < 6 ) where ( p ) is profit in terms of thousands of dollars and ( x ) is the advertising expense in terms of thousands of dollars. what does an ( x )-intercept mean in the context of the problem?

a the number of times the company spent zero dollars on advertising
b the profit when the company spent zero dollars on advertising
c the advertising expense when the company had the most profit
d the advertising expense when the company’s profit was zero dollars

1. copper production increased at a rate of about 4.9% per year between 1988 and 1993. in 1993, copper production was approximately 1.801 billion kilograms. if this trend continued, which equation best models the copper production (( p )) in billions of kilograms, since 1993? (let ( t = 0 ) for 1993.)

a ( p = 1.801(4900)^t )
b ( p = 1.801(1400)^t )
c ( p = 1.801(1.049)^t )
d ( p = 1.801(0.049)^t )

1. the profit ( p ), in dollars, for a company is modeled by the function ( p(x) = -750x^2 + 15000x ) where ( x ) is the number of items produced. for which values of ( x ) will the company lose money?

a ( x < 2 )
b ( 2 < x leq 10 )
c ( 10 leq x < 20 )
d ( x > 20 )

1. which circle has the smallest area?

a ( x^2 + y^2 = 12 )
b ( (x - 2)^2 + y^2 = 8 )
c ( (x + 1)^2 + (y + 3)^2 = 8 )
d ( (x + 8)^2 + (y - 9)^2 = 3 )

1. solve for ( x ): ( \frac{1}{3}|2x + 6| + 2 = 0 )

a ( x = 5 ) or ( x = 1 )
b ( x = 5 )
c ( x = -5 ) or ( x = -1 )
d ( x = -1 )

1. what is the solution set of the system below?

( x = 2y )
( x - y^2 = -2y )
a ( {(0, 0)} )
b ( {(0, 4)} )
c ( {(0, 0), (4, 0)} )
d ( {(0, 0), (8, 4)} )

Explanation:

Question 13

Step1: Define x-intercept meaning

An x-intercept occurs where $P=0$ in the function $P(x)$.

Step2: Map to problem context

Here, $P$ = profit (thousands of dollars), $x$ = advertising expense (thousands of dollars). When $P=0$, profit is zero, and $x$ is the corresponding advertising cost.

Step1: Identify growth rate formula

Exponential growth formula: $P = P_0(1+r)^t$, where $r$ = annual growth rate (decimal).

Step2: Convert percentage to decimal

$4.9\% = 0.049$, so $1+r = 1+0.049 = 1.049$.

Step3: Substitute known values

$P_0=1.801$ (1993 production), so $P = 1.801(1.049)^t$.

Step1: Set profit < 0 (losing money)

$P(x) = -750x^2 + 15000x < 0$

Step2: Factor the quadratic

$750x(-x + 20) < 0$ or $-750x(x-20) < 0$

Step3: Solve inequality

Since $x$ (items produced) > 0, the inequality holds when $x-20 > 0$, so $x>20$.

Answer:

D. the advertising expense when the company's profit was zero dollars

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Question 14