The Sweet Drip Beverage C. sells cans of soda pop in machines. It finds that sales average 26000 cans per month when the cans sell for 50cents each. For each nickel increase in the price, the sales per month drop by 1000 cans.
Determine a function R(x) that models the total revenue realized by Sweet Drip, where x is the number of $0.05 increases in the price of a can.
Sales S (in cans/month)
= 26000 - 1000 x
where P is the price in cents.
Revenue R = S*P = 26,000 -
Sales S (in cans/month)
= 26000 - 1000 x
Price P = 50 + 5x
Monthly Revenue R = S*P
= (26,000 - 1000x)(50 + 5x)
Multiply it out.
To determine the function R(x) that models the total revenue realized by Sweet Drip, we need to consider two factors: the quantity sold and the price.
1. Quantity Sold:
The given information states that sales average 26,000 cans per month when the cans sell for 50 cents each. For each nickel increase in the price, the sales per month drop by 1,000 cans.
Let's denote the quantity of cans sold per month as Q. Since each nickel increase in price reduces the quantity sold by 1,000 cans, we can write the quantity as:
Q = 26,000 - 1,000x
Where x is the number of $0.05 increases in the price of a can.
2. Price:
The price of a can starts at 50 cents and increases by 5 cents for each increment x. The original price is $0.50, and for each increment x, the price increases by $0.05. Therefore, we can determine the price as:
P = 0.50 + 0.05x
The total revenue (R) is given by the formula: R = Q * P
Substituting the equations for Q and P, we get:
R(x) = (26,000 - 1,000x)(0.50 + 0.05x)
Therefore, the function R(x) that models the total revenue realized by Sweet Drip is:
R(x) = (26,000 - 1,000x)(0.50 + 0.05x)
To determine the function R(x) that models the total revenue realized by Sweet Drip, we need to consider the relationship between the price increase and the corresponding change in sales.
Here's how we can approach this:
1. Identify the initial price and sales data point:
- Price of a can: $0.50
- Sales per month: 26,000 cans
2. Determine the change in sales per nickel increase in price:
- The problem states that the sales per month drop by 1000 cans for each nickel increase in price.
Therefore, for every $0.05 increase in price, the sales will decrease by 1000 cans.
3. Calculate the new quantity of sales after price adjustments:
Since sales decrease by 1000 cans for each $0.05 increase in price, we can express this as a linear relationship:
- For each $0.05 increase in price (x), sales decrease by 1000 cans.
So, the formula for the new quantity of sales (S) in terms of the price increase (x) can be written as:
S = 26,000 - 1000x
4. Determine the total revenue:
The total revenue (R) is calculated as the product of the price per can and the quantity of sales.
- Price per can: $0.50 + $0.05x
- Quantity of sales: 26,000 - 1000x
Therefore, the function R(x) can be expressed as:
R(x) = (0.50 + 0.05x) * (26,000 - 1000x)
So, the function R(x) that models the total revenue realized by Sweet Drip is:
R(x) = (0.50 + 0.05x) * (26,000 - 1000x)