Problem 109 - Algebra for College Students

Question

A pizza parlor charges \(\$ 14.95\) for a pizza plus \(\$ 0.50\) for each topping. Express the total cost of a pizza \(C(n)\) in dollars as a function of the number of toppings \(n .\) Find \(C(6)\)

Step-by-Step Solution

Answer

The total cost for a pizza with 6 toppings is \(\$ 17.95\).

Step 1 - Define the function

The total cost of a pizza with toppings can be expressed as a function of the number of toppings. The base price of the pizza is \(\$ 14.95\) and each topping adds \(\$ 0.50\). Therefore, the function for the total cost \(C(n)\) can be written as follows: \[C(n) = 14.95 + 0.50n\]

Step 2 - Substitute the number of toppings

To find the cost of a pizza with 6 toppings, substitute \(n = 6\) into the cost function \(C(n)\): \[C(6) = 14.95 + 0.50(6)\]

Step 3 - Perform the calculation

Now, calculate the value: \[C(6) = 14.95 + 0.50 \times 6 = 14.95 + 3.00 = 17.95\]

Key Concepts

linear equations

Linear equations are equations of the first degree, which means they involve variables raised to the power of one. They look like a straight line when graphed. In the case of the pizza cost problem, the equation we form is linear because it involves a constant term and a variable term that is multiplied by its coefficient. This particular equation is: \(C(n) = 14.95 + 0.50n\). Here, \(C(n)\) is the total cost of a pizza and \(n\) represents the number of toppings. The base price of the pizza is given by the constant 14.95 and for each additional topping, you add 0.50 to the total. Linear equations are essential for understanding relationships where one quantity changes at a constant rate with respect to another.

function notation

Function notation is a way to write equations that describe a relationship between input and output values. In simpler terms, it's a way of showing how changing one value affects another. For the pizza cost problem, we use the notation \(C(n)\). This means that the total cost \(C\) is a function of the number of toppings \(n\). It tells us that you input the number of toppings, and the function gives you the output—the total cost. Using function notation makes it easier to substitute values and compute results, like finding the cost for 6 toppings as \(C(6)\). Essentially, function notation helps in representing the dependent and independent variables in a clear manner.

substitution method

The substitution method involves replacing a variable with a known value to solve an equation. In our exercise, we needed to find the cost of a pizza with 6 toppings. To do this, we substitute \(n = 6\) into the function \(C(n)\). Here's how:

\(C(6) = 14.95 + 0.50 \times 6\).

By substituting 6 for \(n\), we're able to compute the total cost directly. This method is handy for solving equations because it reduces the number of variables, making calculations straightforward. It finds frequent use in algebra for solving systems of equations, evaluating functions, and simplifying mathematical expressions.

algebraic expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operators (like addition or multiplication). They represent quantities in a general form. For example, the expression \(14.95 + 0.50n\) from our pizza exercise includes:

• A constant (14.95, the base price)
• A coefficient (0.50, the cost per topping)
• A variable (n, the number of toppings)

Algebraic expressions summarize complex relationships using concise mathematical terms, allowing us to perform operations, simplify, and solve problems efficiently. In our context, they help us represent and calculate the pizza's total cost depending on the number of toppings added.