a fast food place sells 425 large orders of fries per week at price $2.75 each. A market survey indicates that for each $0.10 decrease, sales will increase by 20 orders of fires
a) Determine the ,marginal revenue function
b) With what price with marginal revenue be equal to zero, interpret this value
let the number of 10 cent decreases by n
then price of fries = 2.75-.1n
number sold = 425 + 20n
revenue = (2.75-.1n)(425+20n)
d(revenue)/dx = (2.75-.1)(20) + (425+20n)(-.1) = 0 for a max of revenue
55 - 2n - 42.5 - 2n = 0
-4n = -12.5
n = 3.125
I would assume that the decreases would be complete multiples of 10 cents, so there should be 3 decreases or a 30 cent decrease
when n = 3
they will have a price of 2.45
and they will sell 485
revenue = 2.45(485) = 1188.25
check:
if n=2
revenue = 2.55(465) = 1185.75 , which is lower
if n=4
revenue = 2.35(505 = 1186.75 , which is also lower than the n=3 value
a) To determine the marginal revenue function, we need to find the rate of change of revenue with respect to the number of orders of fries.
Let's represent the number of $0.10 decreases as "x" and the number of orders of fries as "y." Using the given information, we can write the following equation relating these variables:
Revenue = (425 + 20x) * (2.75 - 0.10x)
The revenue function is the product of the number of orders and the price per order. To find the marginal revenue function, we differentiate this revenue function with respect to x:
Marginal Revenue = d/dx [(425 + 20x) * (2.75 - 0.10x)]
Differentiating gives us:
Marginal Revenue = 20(2.75 - 0.10x) + (425 + 20x)(-0.10)
Simplifying further gives:
Marginal Revenue = 55 - 2x
Therefore, the marginal revenue function is: MR(x) = 55 - 2x.
b) To find the price at which marginal revenue is equal to zero, we set MR(x) equal to zero and solve for x:
55 - 2x = 0
2x = 55
x = 55/2
x ≈ 27.5
The value of x represents the number of $0.10 decreases. Since we cannot have a fractional decrease, x can be rounded up to 28.
Therefore, at a price decrease of $0.10 for every 28 units sold, the marginal revenue will be zero. This means that beyond this point, reducing the price further will not increase revenue.
a) To determine the marginal revenue function, we need to find the rate of change of revenue with respect to the number of orders.
Let's begin by finding the original revenue generated by selling 425 large orders of fries at a price of $2.75 each.
Original Revenue = Number of orders * Price per order
= 425 * $2.75
= $1168.75
Next, we need to consider the increase in sales due to a decrease in price. According to the market survey, for each $0.10 decrease in price, sales increase by 20 orders of fries.
Let's calculate the new number of orders and the corresponding revenue for each $0.10 decrease in price.
Price decrease = $0.10
Number of additional orders = 20
New price = $2.75 - $0.10 = $2.65
New number of orders = 425 + 20 = 445
New Revenue = New number of orders * New price
= 445 * $2.65
= $1179.25
Now we can determine the marginal revenue, which is the difference between the new and original revenues.
Marginal Revenue = New Revenue - Original Revenue
= $1179.25 - $1168.75
= $10.50
Therefore, the marginal revenue function is given by:
MR(x) = $10.50, where x is the number of orders.
b) To find the price at which marginal revenue is equal to zero, we need to find the point at which the marginal revenue function intersects the x-axis.
Since the marginal revenue function is a constant $10.50, it does not intersect the x-axis. Therefore, there is no price at which the marginal revenue is equal to zero.
Interpretation: This means that the company can continue to lower the price to increase sales without reaching a point where the marginal revenue becomes zero. By lowering the price, they can attract more customers and potentially increase their overall profits.
a) To determine the marginal revenue function, we need to find the relationship between the price and the number of orders of fries sold.
Given that the initial price is $2.75 and the quantity of orders is 425, we will start by calculating the revenue at this price. Revenue is calculated by multiplying the price by the quantity: 2.75 * 425 = $1168.75.
Next, we'll consider a decrease in price of $0.10. This decrease in price will result in an increase in orders of fries by 20. So, if the price decreases to $2.65, the new quantity of orders will be 425 + 20 = 445. Revenue at this new price will be 2.65 * 445 = $1181.25.
Now, we can calculate the marginal revenue by subtracting the initial revenue from the new revenue: $1181.25 - $1168.75 = $12.50. Since this calculation shows the change in revenue due to the change in quantity, we divide it by the change in quantity (20): $12.50 / 20 = $0.625.
Therefore, the marginal revenue per additional order of fries is $0.625. To determine the marginal revenue function, we can express it as a function of the quantity of orders, which is q. So, the marginal revenue function is:
MR(q) = $0.625 * q
b) To find the price at which the marginal revenue is equal to zero, we can set the marginal revenue function equal to zero:
MR(q) = $0.625 * q = 0
Dividing both sides of the equation by $0.625, we get:
q = 0 / $0.625
q = 0
This means that when the quantity of orders is zero, the marginal revenue will be zero. However, in the context of the problem, it doesn't make sense to have zero orders, so we need to interpret this value differently.
Interpretation: The value q = 0 represents the theoretical point at which the fast food place would need to offer the fries for free in order to increase sales. At this price, the marginal revenue is zero because although the number of orders would increase, the revenue generated would be zero due to the absence of price per order.