Problem 87 - Algebra for College Students

Question

Solve each problem. The function \(C=0.50 t+8.95\) gives the customer's cost in dollars for a pan pizza, where \(t\) is the number of toppings. a) Find the cost of a five-topping pizza. b) Find \(t\) if \(C=14.45\) and interpret your result.

Step-by-Step Solution

Answer

a) \$11.45; b) 11 toppings

Step 1: Understand the function

The given function is: \[ C = 0.50t + 8.95 \]where \( C \) is the cost in dollars, and \( t \) is the number of toppings.

Step 2: Substitute and solve for part (a)

For part (a), find the cost of a five-topping pizza. Substitute \( t = 5 \) into the function:\[ C = 0.50(5) + 8.95 \]Solve this to get:\[ C = 2.50 + 8.95 = 11.45 \]So, the cost of a five-topping pizza is \$11.45.

Step 3: Set up the equation for part (b)

For part (b), find \( t \) when \( C = 14.45 \). Substitute \( C = 14.45 \) into the function:\[ 14.45 = 0.50t + 8.95 \]

Step 4: Solve for \( t \)

Isolate \( t \) on one side of the equation. Subtract 8.95 from both sides:\[ 14.45 - 8.95 = 0.50t \]This simplifies to:\[ 5.50 = 0.50t \]Next, divide both sides by 0.50:\[ t = \frac{5.50}{0.50} = 11 \]Thus, \( t = 11 \).

Step 5: Interpret the result for part (b)

The result \( t = 11 \) means that if the cost of the pizza is \$14.45, there are 11 toppings on the pizza.

Key Concepts

Cost Function

In mathematics, a cost function is an equation that describes the total cost of producing a specific number of items or completing a certain task. In this case, our cost function is: \[ C = 0.50t + 8.95 \]Here, \( C \) represents the total cost in dollars, and \( t \) stands for the number of toppings on a pizza. Think of each part of the equation as a contributor to the final cost.

• Fixed Costs: The term \( 8.95 \) in our equation is the base cost of the pizza with no toppings.
• Variable Costs: The term \( 0.50t \) increases the total cost based on the number of toppings.

So, the cost equation combines both fixed and variable costs to provide the total cost of a pizza based on the number of toppings.

Solving Linear Equations

Solving linear equations means finding the value of the variable that makes the equation true. Let's see how this works in two parts. First, for part (a): the goal is to find the cost (\( C \)) for a given number of toppings (\( t \)). Here, \( t = 5 \). Substitute \( t = 5 \) into the given function: \[ C = 0.50(5) + 8.95 \] After simplifying, we get: \[ C = 2.50 + 8.95 = 11.45 \] So, the cost of a five-topping pizza is \$11.45.In part (b), we start with the cost (\( C \)) and need to find the number of toppings (\( t \)). Given \( C = 14.45 \), we set up our equation as follows: \[ 14.45 = 0.50t + 8.95 \] Next, isolate the variable \( t \) by subtracting 8.95 from both sides: \[ 14.45 - 8.95 = 0.50t \] This simplifies to: \[ 5.50 = 0.50t \] Finally, divide both sides by 0.50: \[ t = \frac{5.50}{0.50} = 11 \] So, \( t = 11 \), meaning a pizza costing \$14.45 has 11 toppings.

Interpreting Linear Equations

Interpreting linear equations involves understanding what the equation and its terms represent in practical terms. In our problem, the linear equation \( C = 0.50t + 8.95 \) helps us determine the cost of a pizza based on the number of toppings.

• The slope (0.50) indicates the additional cost per topping.
• The y-intercept (8.95) is the base cost of the pizza with no toppings.

When we find that \( t = 11 \) for \( C = 14.45 \), it shows that an 11-topping pizza will cost \$14.45. Interpreting these results helps us understand how each topping affects the total cost, making the equation useful for planning and budgeting.

Variable Substitution

Variable substitution involves replacing variables with specific values to simplify an equation. Take this step-by-step approach to solve:For part (a), let \( t = 5 \) and insert it into the function: \[ C = 0.50(5) + 8.95 \] After simplifying, you find the cost is \$11.45.For part (b), with \( C = 14.45 \), substitute this value into the function to find \( t \): \[ 14.45 = 0.50t + 8.95 \] Isolate \( t \): \[ 14.45 - 8.95 = 0.50t \] This gives you: \[ 5.50 = 0.50t \]Finally, solve for \( t \) by dividing by 0.50: \[ t = 11 \] So, 11 toppings are required to reach a total cost of \$14.45. This simple method allows you to convert verbal problems into mathematical solutions efficiently.