Lesson 8

Exponential Situations as Functions

Problem 1

For an experiment, a scientist designs a can, 20 cm in height, that can hold water. A tube is installed at the bottom of the can allowing water to drain out.

At the beginning of the experiment, the can is full. When the experiment starts, the water begins to drain, and the height of the water in the can decreases by a factor of \(\frac13\) each minute.

  1. Explain why the height of the water in the can is a function of time.
  2. The height, \(h\) , in cm, is a function \(f\) of time \(t\) in minutes since the beginning of the experiment, \(h = f(t)\). Find an expression for \(f(t)\).
  3. Find and record the values for \(f\) when \(t\) is 0, 1, 2, and 3.
  4. Find \(f(4)\). What does \(f(4)\) represent?
  5. Sketch a graph of \(f\) by hand or use graphing technology.
  6. What happens to the level of water in the can as time continues to elapse? How do you see this in the graph?

Solution

For access, consult one of our IM Certified Partners.

Problem 2

A scientist measures the height, \(h\), of a tree each month, and \(m\) is the number of months since the scientist first measured the height of the tree.

  1. Is the height, \(h\), a function of the month, \(m\)? Explain how you know.
  2. Is the month, \(m\), a function of the height, \(h\)? Explain how you know.

Solution

For access, consult one of our IM Certified Partners.

Problem 3

A bacteria population is 10,000. It triples each day.

  1. Explain why the bacteria population, \(b\), is a function of the number of days, \(d\), since it was measured to be 10,000.
  2. Which variable is the independent variable in this situation?
  3. Write an equation relating \(b\) and \(d\).

Solution

For access, consult one of our IM Certified Partners.

Problem 4

  1. Is the position, \(p\), of the minute hand on a clock a function of the time, \(t\)?
  2. Is the time, \(t\), a function of the position of the minute hand on a clock?

Solution

For access, consult one of our IM Certified Partners.

Problem 5

The area covered by a city is 20 square miles. The area grows by a factor of \(1.1\) each year since it was 20 square miles.

  1. Explain why the area, \(a\), covered by the city, in square miles, is a function of \(t\), the number of years since its area was 20 square miles.
  2. Write an equation for \(a\) in terms of \(t\).

Solution

For access, consult one of our IM Certified Partners.

Problem 6

The graph shows an exponential relationship between \(x\) and \(y\).

  1. Write an equation representing this relationship.
  2. What is the value of \(y\) when \(x = \text-1\)? Label this point on the graph.
  3. What is the value of \(y\) when \(x = \text-2\)? Label this point on the graph.
Graph of function on grid.

Solution

For access, consult one of our IM Certified Partners.

(From Unit 5, Lesson 7.)

Problem 7

Here is an inequality: \(3x+1>34-4x\)

Graph the solution set to the inequality on the number line.

Blank horizontal number line from negative 10 to 10 by 1’s.

Solution

For access, consult one of our IM Certified Partners.

(From Unit 2, Lesson 19.)

Problem 8

Here are the equations that define three functions.

\(f(x)=4x-5\)

\(g(x)=4(x-5)\)

\(h(x)=\frac x 4 - 5\)

  1. Which function value is the largest: \(f(100)\), \(g(100)\), or \(h(100)\)?
  2. Which function value is the largest: \(f(\text-100)\), \(g(\text-100)\), or \(h(\text-100)\)?
  3. Which function value is the largest: \(f(\frac{1}{100})\), \(g(\frac{1}{100})\), or \(h(\frac{1}{100})\)?

Solution

For access, consult one of our IM Certified Partners.

(From Unit 4, Lesson 4.)