Museum Trip: Budget Models

Use linear models to estimate costs and explain what their numbers mean.

Read the Cost Model

Use the model for each estimate. Model predictions do not have to equal observed costs exactly.

A class is planning a museum trip. Records from similar trips pair the number of students, n, with the total cost, C, in dollars. A linear model for these data is C = 21n + 450. The model allows $450 for the bus, $9 per student for a meal, and $12 per student for a museum ticket. Adult costs are covered separately. Use this model unless a question states a change.

Records from Similar Trips
Students, n	Observed cost ($)
24	960
30	1092
36	1194
42	1340

What does the slope of 21 mean in this model?

A. Each additional student adds $450 to the predicted total cost.B. The predicted total cost is $21 for any group size.C. Each additional student adds $21 to the predicted total cost.D. Each additional student lowers the predicted total cost by $21.

What does the intercept of 450 represent in this model?

A. It represents the predicted total cost for 450 students.B. It represents the fixed bus cost before student costs are added.C. It represents the predicted meal cost for each student.D. It represents the predicted ticket cost for the whole group.

Use C = 21n + 450 to predict the total cost for 34 students.

Predicted total cost: $

Use the trip records and the model. How much higher was the observed cost for 30 students than the predicted cost?

The observed cost was $ higher.

The planned group increases from 29 students to 35 students. Use the slope to find the increase in predicted total cost.

Predicted increase: $

Make Budget Decisions

Apply the models to each planning decision. Include the fixed bus cost.

The class can spend at most $1,290. According to C = 21n + 450, what is the greatest whole number of students it can afford?

Greatest number: students

For 38 students, a school grant will pay $336 of the predicted total cost. The students will split the remaining cost equally. Use C = 21n + 450.

How much should each student pay? $

The meal estimate rises from $9 to $11 per student. The bus and ticket estimates stay the same. Which revised model should the class use?

A. C = 23n + 450B. C = 32n + 450C. C = 23n + 452D. C = 21n + 452

For the next two questions, compare two trip packages. Each includes a bus, meals, and museum tickets. The equations predict total cost in dollars for n students.

Two Trip Packages
Package	Cost model
Standard	C = 21n + 450
Group deal	C = 18n + 570

For 32 students, how much less does the Standard package predict than the Group deal?

The Standard package predicts $ less.

10.

For what number of students do the two packages predict the same total cost?

The predictions are equal for students.

Interpret and Check

Pay attention to what each variable measures. Explain the final answer using the model.

The class sets aside $1,500 for the Standard package. Using C = 21n + 450, the predicted money remaining after paying for the trip is R = 1050 − 21n, where R is in dollars.

11.

What does the slope of −21 mean in the remaining-money model?

A. Each additional student decreases the predicted trip cost by $21.B. Each additional student decreases the predicted money remaining by $1,050.C. Each additional student increases the predicted money remaining by $21.D. Each additional student decreases the predicted money remaining by $21.

12.

The bus company reduces the fixed bus estimate by $90. The $1,500 budget and the $21 cost per student stay the same. Complete the revised model for money remaining.

R = − 21n

13.

For a new trip, the original model predicts $1,206 for 36 students. The table shows an observed cost of $1,194 for an earlier trip with 36 students. Which conclusion is best?

A. A future cost may differ because $1,206 is a model estimate.B. Every future trip with 36 students must cost exactly $1,206.C. The model cannot be used because one observed cost differs.D. Every future trip with 36 students must cost exactly $1,194.

14.

Using C = 21n + 450, Noor predicts a total cost of $1,080 for 30 students. She divides $1,080 by 30 and gets $36 per student. She says, “The slope should be 36, not 21.” Explain why $36 per student and a slope of 21 can both be correct. Include the bus cost in your explanation.