You are buying packages of beads to make bracelets. Each package costs $3 plus 7% tax. You have a gift card for $15, and you plan to use only the gift card to pay for the beads. Which function c(p) represents the total cost, where p represents the number of packages of beads you purchase? What is a reasonable domain?
c(p) = 3.00p * 1.07
since c <= 15,
3.00p*1.07 <= 15.00
p <= 5/1.07 = 4.67
So, a max of 4 packages can be purchased
The function c(p) that represents the total cost can be calculated by adding the cost of each package of beads to the total tax amount.
To calculate the cost of each package, we add the price ($3) and the tax (7% of $3, which is $0.21). Therefore, the cost of each package is $3 + $0.21 = $3.21.
So the function c(p) = 3.21p represents the total cost, where p is the number of packages purchased.
The domain for this function would be the set of positive integers because you cannot purchase a fraction or a negative number of packages. Therefore, a reasonable domain for this function is p > 0.
To determine the total cost of purchasing packages of beads, we need to consider the cost of each package, including tax. The cost of each package is $3, and tax adds an additional 7% to the cost.
To calculate the cost, we can use the formula:
Cost = (Cost per package) + (Tax on cost per package)
The cost per package is $3, and the tax on the cost per package is 7% of the cost per package.
Tax on cost per package = (7/100) * (Cost per package)
Substituting the values, we get:
Tax on cost per package = (7/100) * (3) = 0.21
Therefore, the total cost function c(p) can be expressed as:
c(p) = p * (Cost per package + Tax on cost per package)
c(p) = p * (3 + 0.21) = p * 3.21
To determine the reasonable domain, we need to consider the constraints. In this case, you have a gift card of $15, and you plan to use only the gift card to pay for the beads.
The maximum number of packages of beads you can purchase is limited by the value of the gift card, as you cannot spend more than $15.
Therefore, the reasonable domain is the set of whole numbers p such that:
p * 3.21 ≤ 15
Dividing both sides of the inequality by 3.21, we get:
p ≤ 15 / 3.21
p ≤ 4.67
Since the number of packages must be a whole number, we round down to the nearest whole number.
Therefore, the reasonable domain for the number of packages is p ≤ 4.
In summary, the total cost function c(p) is c(p) = p * 3.21, and the reasonable domain for the number of packages is p ≤ 4.