Math Problem Statement

Assume the following situation can be modeled by a linear function. Write an equation for the linear function and use it to answer the given question. Be sure you clearly identify the independent and dependent variables. Is a linear model reasonable for the situation​ described? Assume time can be allocated in fractions of a minute.

You can rent time on computers at the local copy center for a ​$6 setup charge and an additional ​$3.00 for every 10 minutes. How much time can be rented for ​$25​?

Question content area bottom Part 1 Select the correct choice below and fill in the answer box to complete your choice.

A. The independent variable is time​ (t), in​ minutes, and the dependent variable is rental cost​ (r), in dollars. The linear function that models this situation is r equals

   enter your response here . ​(Simplify your answer. Do not include the​ $ symbol in your​ answer.)

B. The independent variable is rental cost​ (r), in​ dollars, and the dependent variable is time​ (t), in minutes. The linear function that models this situation is t equals

   enter your response here . ​(Simplify your answer. Do not include the​ $ symbol in your​ answer.)

Part 2    enter your response here minutes can be rented for ​$ 25 25. ​(Round to the nearest minute as​ needed.) Part 3 A linear model ▼   is not is reasonable for this situation.

Solution

Part 1: Model the Situation

We are tasked with modeling the rental cost as a linear function. Let:

  • Independent variable tt: time rented in minutes.
  • Dependent variable rr: rental cost in dollars.

The linear relationship is as follows:

  • Setup charge: $6 (fixed cost).
  • Additional cost: $3.00 for every 10 minutes, equivalent to $0.30 per minute.

The equation for the rental cost is: r=6+0.3tr = 6 + 0.3t where tt is in minutes and rr is in dollars.

Thus, the correct choice is: A. The independent variable is time (tt), and the dependent variable is rental cost (rr). The linear function is: r=6+0.3tr = 6 + 0.3t

Part 2: Time Rented for $25

To determine how much time can be rented for $25, substitute r=25r = 25 into the equation and solve for tt: 25=6+0.3t25 = 6 + 0.3t 256=0.3t25 - 6 = 0.3t 19=0.3t19 = 0.3t t=190.3=63.33 minutes.t = \frac{19}{0.3} = 63.33 \text{ minutes.}

Rounded to the nearest minute, the time is: t=63 minutes.t = 63 \text{ minutes.}

Part 3: Reasonableness of the Linear Model

A linear model is reasonable for this situation because the cost increases at a constant rate of $0.30 per minute after the setup charge. There are no complicating factors such as discounts or nonlinear pricing.

Final Answers:

  1. The linear function is r=6+0.3tr = 6 + 0.3t.
  2. 63 minutes can be rented for $25.
  3. A linear model is reasonable for this situation.

Follow-Up Questions:

  1. What would be the rental cost for 90 minutes of computer time?
  2. If the setup charge increases to $8, how does the linear model change?
  3. How does the slope of the function represent the cost per minute?
  4. What is the domain of the linear function in this context?
  5. How would the solution change if fractional minutes were not allowed?

Tip: Always verify units (minutes, dollars) to ensure consistency when solving linear models!

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Algebra
Modeling Real-World Scenarios

Formulas

Linear equation: y = mx + b

Theorems

Slope-Intercept Form of a Linear Equation

Suitable Grade Level

Grades 7-9