Question

an advertising agency has found that when it promotes a new product in a city the weekly rate of change r of the number of people who are aware of the promotion follows a linear pattern. the weekly rate of change r of people who are aware of the promotion x weeks after it is introduced is described by the equation r = 7200 - 80x. find the x - and r - intercepts and then graph the function on a viewing window that is meaningful in this application. what is the x - intercept? 90 (type an integer or a fraction ) what is the r - intercept? 7200 (type an integer or a fraction ) choose the most appropriate viewing window boundaries for the function a. - 90,90 by - 7200,7200 b. 0,7120 by 80,7200 c. 0,90 by 0,7200 d. 0,80 by 7120,7200

Explanation:

For x - intercept:

Step1: Recall x - intercept definition

The x - intercept is the value of \(x\) when \(R = 0\). So we set \(R=0\) in the equation \(R = 7200-80x\).

$$0=7200 - 80x$$

Step2: Solve for x

Add \(80x\) to both sides of the equation:

$$80x=7200$$

Divide both sides by 80:

$$x=\frac{7200}{80}=90$$

For R - intercept:

Step1: Recall R - intercept definition

The R - intercept is the value of \(R\) when \(x = 0\). So we set \(x = 0\) in the equation \(R=7200 - 80x\).

$$R=7200-80\times0$$

Step2: Simplify the expression

$$R = 7200$$

For viewing window:

• The variable \(x\) represents the number of weeks after the promotion is introduced. So \(x\geq0\). Also, from the x - intercept, we know that the rate of change \(R\) becomes 0 at \(x = 90\) weeks. So the meaningful range for \(x\) is \(0\leq x\leq90\).
• The variable \(R\) represents the weekly rate of change of the number of people aware of the promotion. Since the number of people aware or the rate of change should be non - negative (we can't have a negative number of people or a negative rate of change in this context), and the maximum value of \(R\) occurs when \(x = 0\) ( \(R = 7200\)) and it decreases to 0 at \(x=90\). So the meaningful range for \(R\) is \(0\leq R\leq7200\).

Looking at the options:

• Option A: Negative values for \(x\) and \(R\) are not meaningful as \(x\) (weeks) and \(R\) (rate of change of people) can't be negative.
• Option B: The upper limit of \(x = 7120\) is incorrect as we know \(x\) should be at most 90.
• Option C: \(x\) ranges from 0 to 90 and \(R\) ranges from 0 to 7200, which is consistent with our analysis.
• Option D: The range of \(x\) (0 to 80) and \(R\) (7120 to 7200) is not correct as \(x\) can go up to 90 and \(R\) can go down to 0.

Answer:

s:

• x - intercept: \(\boldsymbol{90}\)
• R - intercept: \(\boldsymbol{7200}\)
• Viewing window: \(\boldsymbol{C. [0,90] \text{ by } [0,7200]}\)