Question

The sales of a cell phone screen protector (in thousands of units) after x hundred dollars is spent on advertising are modeled by s=20(1−e ^kx). When $1500 is spent on advertising and 5000 units are sold:

a. k>0
b. k<0
c k=0
d. k cannot be determined.

Answer 1

When $1500 is spent on advertising, resulting in the sale of 5000 units, we must have k < 0 to satisfy the given sales model equation.

Step-by-step explanation:

To determine whether k is greater than, less than, or equal to zero, we can utilize the information provided that when $1500 is spent on advertising, 5000 units are sold. Given the equation s=20(1-ekx), where s is the sales in thousands of units and x is the hundreds of dollars spent on advertising, we can substitute the known values into the equation.

First, translating the advertising dollars and units sold into the form used by our equation, that becomes x = 15 (since $1500 is '15' in hundreds of dollars) and s = 5 (since 5000 units are '5' in thousands of units).

Substituting these into the equation gives us 5 = 20(1 - e15k). Simplifying, we get 1/4 = 1 - e15k, leading to e15k = 3/4. For this equation to hold true, the exponential component must be positive, as the exponential function, ex, is always positive.

Therefore, to result in a value less than one, which is required to get a fraction (3/4 in this case), k must be negative. Having a negative exponent decreases the value of the exponential function, bringing it below one, which matches our requirement. Hence, option (b), k < 0, is the correct answer.