A manufacturer has been selling 1,200 television sets a week at $400 each. A market survey indicates that for each $30 rebate offered to the buyer, the number of sets sold will increase by 60 per week. Find the demand function.

Question

Quizplus · Accepted Answer

Let x be the number of rebates offered and y be the number of TV sets sold in a week.

Based on the given information:

- Without rebates, 1,200 TV sets are sold per week at $400 each, so the total revenue is 1,200 x $400 = $480,000.
- For each $30 rebate offered, the number of TV sets sold increases by 60 per week. This can be written as:

y = 1,200 + 60x

- However, the price also changes with the rebate, so the revenue function needs to be adjusted. The price per TV set becomes:

$400 - $30x

So the revenue function is:

R(x) = (1,200 + 60x)($400 - $30x)

Simplifying this expression:

R(x) = $480,000 + $24,000x - $18,000x^2

This is a quadratic function with a negative leading coefficient, which means it has a maximum value. To find that maximum value, we need to use the formula:

x = -b/2a

where a = -18,000 and b = 24,000.

Plugging in these values:

x = -24,000 / 2(-18,000) = 0.6667 (rounded to four decimal places)

This means that the maximum revenue occurs when approximately 0.6667 rebates are offered. To find the corresponding maximum revenue, we can plug this value back into the revenue function:

R(0.6667) = $481,777.78

Therefore, the demand function is the equation for y given above:

y = 1,200 + 60x

where x is the number of rebates offered.

A Manufacturer Has Been Selling 1,200 Television Sets a Week