You run a small furniture business. You sign a deal with a customer to deliver up to 500 chairs, the exact number to be determined by the customer later. The price will be $120 per chair up to 400 chairs, and above 400 the price will be reduced by $0.25 per chair (on the whole order) for every additional chair over 400 ordered. What is the largest revenue your company can make under this deal?
I've found the minimum to be zero for this problem because they could sell no chairs at all and from what I can deduct from the problem the maximum they can make should be selling 400 chairs at $120 each which would be $48000 but my online turn in keeps telling me that's wrong and I don't understand why that is, please help.
let the number of chairs over 400 be x
now:
number of chairs = 400
price per chair = 120
Revenue = 48000
after production increase:
number of chairs = 400+x
price per chair = 120 - .25x
Revenue
= R
= (400+x)(120-.25x)
= 48000 + 20x - .25x^2
which is a downwards opening parabola
the x of the vertex is -20/-.5 = 40
so if x = 40
R = 440(120-.25(40)) = $48,400
To determine the maximum revenue your company can make under this deal, we need to consider the different scenarios based on the number of chairs ordered by the customer.
Scenario 1: No chairs ordered
In this scenario, the revenue would be zero since no chairs are being ordered.
Scenario 2: Number of chairs ordered is between 0 and 400
In this range, the price per chair is $120. To maximize revenue, we assume the customer orders the maximum of 400 chairs. Therefore, the revenue in this scenario would be 400 chairs × $120/chair = $48,000.
Scenario 3: Number of chairs ordered is above 400
In this case, the price per chair will be reduced by $0.25 for every additional chair over 400. To maximize revenue, we assume the customer orders the maximum allowed number of chairs, which is 500.
For the first 400 chairs, the price per chair is $120.
For the remaining 100 chairs (500 - 400), the price reduction is $0.25 per chair for the entire order. Therefore, the reduced price per chair is $120 - $0.25 = $119.75.
The revenue for the first 400 chairs is 400 chairs × $120/chair = $48,000.
The revenue for the remaining 100 chairs is 100 chairs × $119.75/chair = $11,975.
Total revenue for this scenario would be $48,000 + $11,975 = $59,975.
Therefore, the largest revenue your company can make under this deal is $59,975.
To find the largest revenue your company can make under this deal, we need to consider the pricing structure based on the number of chairs ordered.
First, let's calculate the revenue for up to 400 chairs. The price per chair is $120, so the revenue for 400 chairs would be 400 * $120 = $48,000. This matches the value you obtained, so this is correct.
However, the problem states that the price per chair will be reduced by $0.25 for every additional chair over 400. To determine the largest revenue, we need to find the point where the reduction in price per chair negates the increase in the number of chairs.
Let's calculate the reduction in revenue when the number of chairs is above 400. The price reduction is $0.25 per chair for every additional chair over 400. So, for each chair over 400, the reduced revenue per chair is $0.25 less than $120.
To find the point at which the revenue starts decreasing, we set up an equation:
$120 - $0.25 * (X - 400) = $120
Here, X represents the number of chairs. We subtract (X - 400) because we want to find the number of chairs above 400. Solving this equation gives us:
$120 - $0.25X + $100 = $120
Simplifying the equation gives us:
$100 - $0.25X = 0
$0.25X = $100
X = $100 / $0.25
X = 400
This means that when the number of chairs exceeds 400, the revenue starts decreasing. Therefore, the largest revenue your company can make is selling 400 chairs at $120 each, which is $48,000.
If your online turn-in system is not accepting this answer, you may want to double-check the problem statement or seek clarification from your instructor to ensure there are no additional constraints or conditions.