Question

a company finds that if it charges x dollars for a cell phone, it can expect to sell 1000 - 2x phones. the company uses the function r defined by r(x) = x(1000 - 2x) to model the expected revenue, in dollars, from selling the cell phones at x dollars each. is 0 ≤ x ≤ 600 an appropriate domain for function r, assuming the company wants to make money? explain your reasoning.

Explanation:

Step1: Analyze Revenue Function

The revenue function is \( r(x) = x(1000 - 2x) \). For the company to make money, \( r(x)>0 \). Also, the number of phones sold \( 1000 - 2x \) must be non - negative (you can't sell a negative number of phones), so \( 1000 - 2x\geq0\Rightarrow x\leq500 \). And the price \( x\geq0 \) (you can't charge a negative price).

Step2: Solve \( r(x)>0 \)

Set \( r(x)=x(1000 - 2x)>0 \). The solutions to the equation \( x(1000 - 2x)=0 \) are \( x = 0 \) and \( x=\frac{1000}{2}=500 \). Since the function \( y = r(x) \) is a quadratic function with \( a=- 2<0 \) (the coefficient of \( x^{2} \) when we expand \( r(x)=-2x^{2}+1000x \)), the parabola opens downwards. So \( r(x)>0 \) when \( 0 < x < 500 \). But we also need to consider the domain where the number of phones sold is non - negative. When \( x = 0 \), \( r(0)=0 \) (no revenue), and when \( x = 500 \), \( r(500)=500\times(1000 - 2\times500)=0 \). The domain \( 0\leq x\leq500 \) includes the values of \( x \) where \( r(x)\geq0 \), but for making money (positive revenue), we need \( 0 < x < 500 \). However, the domain \( 0\leq x\leq500 \) is appropriate in the sense that it is the set of all non - negative prices where the number of phones sold is non - negative. If we consider the domain for which the company can potentially make money (excluding the endpoints where revenue is zero), but the question is about whether \( 0\leq x\leq500 \) is appropriate for the function \( r \) (not just for positive revenue). Since \( x\) (price) can't be negative and the number of phones sold \( 1000 - 2x \) can't be negative, the domain of \( x \) should satisfy \( 0\leq x\leq500 \).

Answer:

Yes, \( 0\leq x\leq500 \) is an appropriate domain. The price \( x \) must be non - negative (\( x\geq0 \)) as you can't charge a negative price. The number of cell phones sold is \( 1000 - 2x \), and this must be non - negative (you can't sell a negative number of phones), so \( 1000 - 2x\geq0\Rightarrow x\leq500 \). The function \( r(x)=x(1000 - 2x) \) is defined for all \( x \) in \( 0\leq x\leq500 \), and this domain represents all valid price points (from a price of \( 0 \) dollars up to the price where no phones are sold, \( 500 \) dollars) for the revenue function.