On the basis of past experience, a company has determined that its sales revenue (in dollars) is related to its advertising according to the formula , where x is the amount spent on radio advertising and y is the amount spent on television advertising. If the company plans to spend $30,000 on these two means of advertising, how much should it spend on each method to maximize its sales revenue?
A) $8,401.47 on radio; $21,598.53 on television
B) $10,801.89 on radio; $19,198.11 on television
C) $12,002.10 on radio; $17,997.90 on television
D) $14,402.52 on radio; $15,597.48 on television
E) $15,602.73 on radio; $14,397.27 on television

Question

On the basis of past experience, a company has determined that its sales revenue (in dollars) is related to its advertising according to the formula  , where x is the amount spent on radio advertising and y is the amount spent on television advertising. If the company plans to spend $30,000 on these two means of advertising, how much should it spend on each method to maximize its sales revenue? ​
A) $8,401.47 on radio; $21,598.53 on television 
B) $10,801.89 on radio; $19,198.11 on television 
C) $12,002.10 on radio; $17,997.90 on television 
D) $14,402.52 on radio; $15,597.48 on television 
E) $15,602.73 on radio; $14,397.27 on television

Quizplus · Accepted Answer

To find the optimal spending on each method, we need to maximize the sales revenue formula in terms of x and y. We can start by isolating y:

y = (30,000 - x) / 2

Then we substitute this expression for y in the sales revenue formula:

R(x) = 20,000 + 9x - 0.002x^2 - 0.005(30,000 - x)^2

Simplifying and expanding:

R(x) = 20,000 + 9x - 6x^2/1000 - 225 - 30x + x^2/200

R(x) = -6x^2/1000 + (9 + 1/200)x - 205

To find the maximum, we take the derivative and set it to zero:

R'(x) = -12x/1000 + 9/200 = 0

Solving for x:

x = 12,002.1

Substituting back into our expression for y:

y = (30,000 - 12,002.1) / 2 = 17,997.9

Therefore, the optimal spending is $12,002.10 on radio and $17,997.90 on television, so the answer is C.

On the Basis of Past Experience, a Company Has Determined