Math Problem Statement

The number of drive-in movie theaters in the United States from 1995 to 2018 can be modeled by the function $d(t)=-0.086t^3+3.71t^2-53.7t+643$ where $t$ is the number of years after 1995.

a. Use technology to identify the graph of the function for $1\le t\le23$ .

Responses

Graph of a curve on a coordinate plane. The curve begins at ordered pair 1 comma 450, passes through the ordered pairs 4 comma 550, 8 comma 600 and ends at ordered pair 14 comma 610.

  • image with description: Graph of a curve on a coordinate plane. The curve begins at ordered pair 1 comma 450, passes through the ordered pairs 4 comma 550, 8 comma 600 and ends at ordered pair 14 comma 610. - - no response given

Graph of a curve on a coordinate plane. The curve begins at ordered pair 1 comma 600, passes through the ordered pairs 8 comma 400, 12 comma 390 and ends at ordered pair 22 comma 340.

  • image with description: Graph of a curve on a coordinate plane. The curve begins at ordered pair 1 comma 600, passes through the ordered pairs 8 comma 400, 12 comma 390 and ends at ordered pair 22 comma 340. - - correct

Graph of a curve on a coordinate plane. The curve begins at ordered pair 0 comma 200, passes through the ordered pairs 8 comma 400, 12 comma 400 and ends at ordered pair 16 comma 400.

  • image with description: Graph of a curve on a coordinate plane. The curve begins at ordered pair 0 comma 200, passes through the ordered pairs 8 comma 400, 12 comma 400 and ends at ordered pair 16 comma 400. - - no response given

Graph of a curve on a coordinate plane. The curve begins at ordered pair 2 comma 800, passes through the ordered pairs 8 comma 800, 12 comma 400 and ends at ordered pair 16 comma 200.

  • image with description: Graph of a curve on a coordinate plane. The curve begins at ordered pair 2 comma 800, passes through the ordered pairs 8 comma 800, 12 comma 400 and ends at ordered pair 16 comma 200. - - no response given Question 2 Describe the behavior of the graph on this interval. 0 / 10000 Word Limit Question 3 b. Find and interpret the average rates of change in the number of drive-in movie theaters from 1996 to 2006 and from 2006 to 2018. Round your answers to the nearest whole number. From 1996 to 2006 the average rate of change is about theaters per year.

From 2006 to 2018 the average rate of change is about theaters per year.

Question 4 c. Do you think this model can be used for years before 1995 or after 2018? Responses

Yes Yes - no response given

No No - correct Question 5 Explain. 0 / 10000 Word Limit Skip to navigation

Solution

Here is the graph of the number of drive-in movie theaters in the U.S. from 1996 to 2018, modeled by d(t)=−0.086t3+3.71t2−53.7t+643d(t) = -0.086t^3 + 3.71t^2 - 53.7t + 643, where tt is the number of years after 1995 (i.e., 1≤t≤231 \leq t \leq 23).

For part (b): We are asked to find and interpret the average rates of change of drive-in theaters from 1996 to 2006 and from 2006 to 2018.

  1. Average rate of change from 1996 to 2006: Average Rate of Change=d(11)−d(1)11−1\text{Average Rate of Change} = \frac{d(11) - d(1)}{11 - 1}
  2. Average rate of change from 2006 to 2018: Average Rate of Change=d(23)−d(11)23−11\text{Average Rate of Change} = \frac{d(23) - d(11)}{23 - 11}

I'll calculate these next.The average rates of change in the number of drive-in movie theaters are:

  • From 1996 to 2006: about −21-21 theaters per year (indicating a decline).
  • From 2006 to 2018: about −5-5 theaters per year (indicating a slower decline).

For part (c): The model should not be used for years before 1995 or after 2018 because it was specifically designed to fit the data between those years. Extrapolating outside this range could lead to inaccurate predictions, as the behavior of the model may not reflect real-world trends beyond 1995–2018.

Would you like more details or have any questions?

Here are 5 related questions:

  1. How is the average rate of change related to the slope of the function?
  2. What does the negative average rate of change imply about the number of drive-ins?
  3. How can we estimate the number of drive-ins for a specific year using the model?
  4. What might cause the decline in the number of drive-in theaters?
  5. How does a cubic function like this behave over long time periods?

Tip: Average rate of change is simply the slope of the secant line between two points on the curve.

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Math Problem Analysis

Mathematical Concepts

Algebra
Cubic Functions
Rates of Change

Formulas

d(t) = -0.086t^3 + 3.71t^2 - 53.7t + 643
Average Rate of Change = (f(b) - f(a)) / (b - a)

Theorems

Secant Line Slope
Cubic Function Behavior

Suitable Grade Level

Grades 10-12

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