"Three hundred books sell for $40 each, resulting in a revenue of $12,000. For each $5 increase in price, 25 fewer books are sold. Write revenue R as a function of the number x of $5 increases." How would you get this problem started?
Notice that the decrease in books numbers is always 5 times the increase in price so
R= book numbers * price
In this case books numbers is (300-25) which is (300-5x) ,and price is (40+5) which is (40+x)
R = (300-5x)(40+x)
= 12000-200x+300x-5(x^2)
=12000+100X -5(X^2)
To check if you consider there is no increase in price x=0 which means R will be 12000$ as in question and if u consider it is 5 it will give u R=12375$ which is the new R( 275*45)
Well, this problem seems to involve both algebra and economics. Quite the combo! Let's break it down step by step. To get started, we need to find a way to express revenue (R) as a function of the number of $5 increases (x).
First, we know that initially, the price of each book is $40 and 300 books are sold, resulting in a revenue of $12,000. So, we can start by defining the initial revenue as a base value.
Let's call the initial revenue when the price is $40 and 300 books are sold as R₀ (pronounced R naught, just to throw some scientific flair). So, we have:
R₀ = $12,000
Now, we're told that for each $5 increase in price, 25 fewer books are sold. This means that the price per book is going to change as we increase the price.
To figure out how much the price changes for each increase, we need to figure out the relationship between the number of increases (x) and the price change. Since each increase is $5, we can say that the total price increase is 5x.
Next, we need to figure out how many books are being sold when the price increases. We know that for each $5 increase, there are 25 fewer books sold. So, we can say that the number of books sold is decreasing by 25x.
Now, let's put it all together. The new revenue (R) can be written as:
R = R₀ - (price decrease) x (books sold decrease)
Substituting the given values, we get:
R = $12,000 - (5x) x (25x)
And there you have it! We've expressed the revenue (R) as a function of the number of $5 increases (x). Now, we can simplify and solve for R to see how revenue changes with each increase in price.
To get started, let's break down the given information and identify the key variables and relationships.
We are told that the initial price of a book is $40, and initially, 300 books are sold resulting in a revenue of $12,000. We are also given the relationship that for every $5 increase in price, 25 fewer books are sold.
Let's define the variables:
- Initial price of a book: p = $40
- Initial number of books sold: n = 300
- Initial revenue: R₀ = $12,000
- Number of $5 increases in price: x
Now, let's determine the relationship between the price increase and the change in the number of books sold. We know that for every $5 increase in price, 25 fewer books are sold.
Based on this information, we can say that:
- For each $5 increase, the number of books sold decreases by 25.
Next, we need to express the revenue function R as a function of the number of $5 increases (x). To do this, we can determine the revenue for each value of x.
Step 1: Calculate the new price for each value of x:
- New price for x increases: pₓ = p + 5x (since each $5 increase in price corresponds to x increases)
Step 2: Calculate the new number of books sold based on the price increase:
- New number of books sold for x increases: nₓ = n - 25x (since for each $5 increase, the number of books sold decreases by 25)
Step 3: Determine the new revenue for x increases:
- New revenue for x increases: Rₓ = pₓ * nₓ
By substituting the expressions for pₓ and nₓ, we can write the revenue function R as a function of the number of $5 increases (x).
To get started, let's break down the given information and try to find relationships between different variables.
1. We are told that 300 books are sold for $40 each, resulting in a total revenue of $12,000. This information allows us to determine the initial revenue and the number of books sold.
2. We are also given that for every $5 increase in price, 25 fewer books are sold. This indicates that the number of books sold is inversely related to the price increase.
Now, let's define the variables:
R: Revenue (in dollars)
x: Number of $5 increases in price (also referred to as the number of price increases)
P: Initial price per book ($40)
N: Initial number of books sold (300)
B: Number of books sold for each $5 increase in price
Next, let's establish the relationships between these variables as given:
1. Revenue can be calculated by multiplying the price per book by the number of books sold:
R = P * N
2. For each $5 increase in price, there is a decrease of 25 books sold. This means that the number of books sold, B, is related to x:
B = N - 25x
Finally, we need to express the revenue, R, in terms of the number of price increases (x). To do this, we substitute the value of B into the equation for R:
R = P * (N - 25x)
Simplifying this equation will give us the revenue as a function of the number of price increases.