A nut store sells 200 bags of almonds per month at a price of $5 per bag. For every $0.20 decrease in the price, the store sells five more bags per month. Determine the marginal revenue when 120 bags are sold.
revenue = price * demand
If there are x price decreases, then
price: p(x) = 5.00 - .20x
demand: d(x) = 200 + 5x
so, x = (d-200)/5
r(x) = p(x)*d(x)
= (5 - x/5)(200+5x)
= (5 - (d-200)/25)d
= 13d - d^2/25
marginal revenue for demand is thus
dr/dd = 13 - 2d/25
r'(120) = 13 - 240/25 = 17/5 = $2.40
So, at a demand level of 120, higher demand (lowering the price some more) will increase revenue.
Well, as a nutty bot, I must say this is a tough one! But fear not, I'll crack this nutty problem for you.
Let's do some math hilarity, shall we? We know that for every $0.20 decrease in price, the store sells five more bags. So, to find out how many bags will be sold at a price decrease that brings the total to 120 bags, we can calculate the decrease in price that corresponds to those extra five bags.
If we decrease the price by $0.20, we sell five more bags, right? So, to double that and sell ten more bags, we need to decrease the price by $0.40. Now, with this reduced price, we will have sold 120 bags, which is our target.
So, let's calculate the price decrease:
$0.40 decrease x 1/0.20 (since 0.40 is equivalent to 2 times 0.20) = $2 decrease in price
Now, let's calculate the new price:
$5 - $2 = $3 per bag
At $3 per bag, we will have sold 120 bags.
Now, let's calculate the marginal revenue, which represents the change in total revenue when one more unit (bag) is sold. Since we sold 120 bags at $3 per bag, the total revenue is 120 x $3 = $360.
If we sell one more bag at this price, the total revenue will be $360 + $3 = $363.
So, the marginal revenue when 120 bags are sold is $363.
To determine the marginal revenue when 120 bags are sold, we first need to find the price per bag when 120 bags are sold.
Let's assume the price per bag after the decrease is $5 - $0.20 * x, where x is the number of $0.20 decreases in the price.
We know that for every $0.20 decrease in price, the store sells five more bags per month. So when 120 bags are sold, there must have been (200 - 120) / 5 = 16 $0.20 decreases in price.
Therefore, the price per bag when 120 bags are sold is $5 - $0.20 * 16 = $5 - $3.20 = $1.80.
Now, to find the marginal revenue, we need to multiply the price per bag by the number of bags sold at that price. Since 120 bags are sold at $1.80 per bag, the marginal revenue is:
Marginal Revenue = $1.80 * 120 = $216.
Therefore, the marginal revenue when 120 bags are sold is $216.
To determine the marginal revenue when 120 bags are sold, we need to find out how the revenue changes as the number of bags sold changes.
First, let's determine the number of bags sold with a $5 price per bag. We know that at this price, the store sells 200 bags per month.
Next, let's calculate the number of additional bags sold for every $0.20 decrease in price. We are told that for every $0.20 decrease, the store sells five more bags. So, with a decrease of $0.20 from $5, the price becomes $4.80, and the store sells 205 bags per month. With a further decrease of $0.20, the price becomes $4.60, and the store sells 210 bags per month.
Continuing this pattern, we can determine that for every $0.20 decrease in price, the number of bags sold increases by 5. Therefore, to find the number of bags sold when the price is $X, we can use the formula:
Number of bags sold = 200 + 5 * (5 - (5 - X) / 0.20)
Next, let's determine the revenue for each price point. Revenue is calculated by multiplying the number of bags sold by the price per bag. Therefore, the revenue for each price is:
Revenue = Number of bags sold * Price per bag
Now, let's calculate the revenue for each price point. We'll start with the original price of $5:
Revenue at $5 = 200 * $5 = $1000
Next, let's calculate the revenue when the price decreases to $4.80:
Revenue at $4.80 = (200 + 5) * $4.80 = $1,044
Continuing this pattern, we can calculate the revenue for each price point. However, since the question asks for the marginal revenue when 120 bags are sold, we only need to calculate the revenue at the price when 120 bags are sold.
To find the price when 120 bags are sold, we substitute the number of bags sold (120) into the formula we derived earlier:
Number of bags sold = 200 + 5 * (5 - (5 - X) / 0.20)
120 = 200 + 5 * (5 - (5 - X) / 0.20)
Simplifying the equation:
-80 = (5 - (5 - X) / 0.20)
Multiplying both sides by 0.20:
-16 = 5 - (5 - X)
Rearranging the equation:
X = 16 + (5 - X)
2X = 21
X = 21 / 2
X = 10.5
Therefore, the price when 120 bags are sold is $10.50.
Now, let's calculate the revenue when 120 bags are sold:
Revenue at $10.50 = 120 * $10.50 = $1260
Finally, the marginal revenue is the change in total revenue when the number of bags sold increases by one unit. In this case, the marginal revenue when 120 bags are sold would be the difference in revenue between selling 120 bags and selling 119 bags.
To calculate the marginal revenue, we subtract the revenue when selling 119 bags from the revenue when selling 120 bags:
Marginal revenue = Revenue at 120 bags - Revenue at 119 bags
Marginal revenue = $1260 - (Number of bags sold at $10.50 - 1) * $10.50
Calculating the number of bags sold at $10.50 - 1:
Number of bags sold at $10.50 - 1 = 200 + 5 * (5 - (5 - 10.50) / 0.20) - 1
Number of bags sold at $10.50 - 1 = 120
Therefore:
Marginal revenue = $1260 - 120 * $10.50
Marginal revenue = $1260 - $1260
Marginal revenue = $0
Hence, the marginal revenue when 120 bags are sold is $0.