End Speedback Question 18

Question

A company wants to identify where to spend money in marketing their new product. They have three cities that they can market in.

 

City A’s revenue for each dollar spent in marketing is represented by the following table:

 

Marketing Cost                Revenue

(in thousands)                 (in thousands)

1                                     23

2                                     28.5

3                                     30

4                                     27.5

5                                     21

6                                     10.5

 

City B’s revenue for each dollar spent in marketing is represented by the equation R(d)= -2d²+11d+6, where R is the revenue earned (in thousands) and d is the amount of dollars spent (in thousands).

 

City C’s revenue for each dollar spent in marketing is represented by the following graph:

 

Which of the three cities should they select?
City A
City C
City B

 

 

 

 

Answer 1

To determine which city the company should choose, we need to find the maximum possible revenue for each city and compare the results.

Let's examine the information provided.

Step 1: Find the Maximum Revenue for City A

City A's revenue is given in a table:

Marketing Cost ($1,000s)Revenue ($1,000s)
1	23
2	28.5
3	30
4	27.5
5	21
6	10.5

Look at the revenue column.

The highest value is 30, which occurs when the marketing cost is 3.

Remember that both amounts are expressed in thousands of dollars.

Therefore:

City A's maximum revenue = $30,000

This occurs when the company spends $3,000 on marketing.

Step 2: Find the Maximum Revenue for City B

City B's revenue is represented by:

R(d) = -2d² + 11d + 6

This is a quadratic equation in the form:

R(d) = ad² + bd + c

Where:

a = -2

b = 11

c = 6

Because the coefficient of d² is negative, the parabola opens downward.

That means its vertex represents the maximum revenue.

The formula for the horizontal coordinate of the vertex is:

d = -b/(2a)

Substitute:

d = -11/[2(-2)]

d = -11/(-4)

d = 2.75

Therefore, City B reaches its maximum revenue when the company spends $2,750 on marketing.

Now substitute d = 2.75 into the original equation:

R(2.75) = -2(2.75)² + 11(2.75) + 6

First, square 2.75:

(2.75)² = 7.5625

Then multiply:

-2(7.5625) = -15.125

11(2.75) = 30.25

Now add everything:

R(2.75) = -15.125 + 30.25 + 6

R(2.75) = 21.125

Since revenue is measured in thousands:

City B's maximum revenue = $21,125

Step 3: Find the Maximum Revenue for City C

City C's information is supposed to be provided in a graph.

To find its maximum revenue, look for the highest point on the graph.

If the graph shows a downward-opening parabola, the highest point is its vertex.

Read the y-coordinate of that point to determine the maximum revenue.

Remember to check whether the graph measures revenue in thousands of dollars.

Unfortunately, City C's graph is not included in the question shown here, so its maximum revenue cannot be determined from the available information.

Step 4: Compare the Cities

So far, we have:

City A: Maximum revenue = $30,000

City B: Maximum revenue = $21,125

City C: Maximum revenue = Unknown without the graph

Comparing A and B:

$30,000 - $21,125 = $8,875

Therefore, City A produces $8,875 more maximum revenue than City B.

Final Answer

Between Cities A and B, City A is the better choice because its maximum revenue is $30,000, compared with $21,125 for City B.

However, we cannot determine the best city overall without examining City C's graph.

If City C's maximum revenue is:

1. Greater than $30,000: Choose City C.
2. Less than $30,000: Choose City A.
3. Exactly $30,000: Cities A and C tie in maximum revenue.

Important Note: This comparison assumes the objective is to maximize total revenue, as suggested by the question. If the objective were instead to maximize profit, we would need to subtract marketing costs from revenue before comparing the cities.

Key Takeaway: When comparing a table, an equation, and a graph, find the maximum output for each representation. For a downward-opening quadratic, use the vertex formula d = -b/(2a). For a table, identify the largest revenue value. For a graph, identify its highest point.