Worksheet: Application of Graphs of Rational Functions

In this worksheet, we practice graphing rational functions, finding the zeros, and determining the types of asymptotes the graph of a rational function can have.

Q1:

Consider the square prism shown in the diagram.

Write its surface-area-to-volume ratio in terms of 𝑥 . Give your answer in standard form.

  • A 4 𝑥 + 8 𝑥 + 5 𝑥 + 1 2 0 𝑥 + 2 2 𝑥 + 6
  • B 4 𝑥 + 8 𝑥 + 5 𝑥 + 1 1 6 𝑥 + 2 0 𝑥 + 6
  • C 2 0 𝑥 + 2 2 𝑥 + 6 4 𝑥 + 8 𝑥 + 5 𝑥 + 1
  • D 1 6 𝑥 + 2 0 𝑥 + 6 4 𝑥 + 8 𝑥 + 5 𝑥 + 1
  • E 1 2 𝑥 + 1 8 𝑥 + 6 4 𝑥 + 8 𝑥 + 5 𝑥 + 1

The diagram shows the graph of the surface-area-to-volume ratio of the prism as a function of 𝑥 . Which of the following is an approximate value of 𝑥 for which the surface-area-to-volume ratio is 1?

  • A3.3
  • B1.5
  • C1.3
  • D2.3
  • E6

Q2:

A team of scientists have been working on the growth of metal oxide nanowires, that is, metal oxide in the form of wires (cylinders) with dimensions in the order of nanometers. They observed that when the nanowires had reached a critical size, namely, a diameter of 50 nm and a length of 250 nm, the diameter increased at a rate of 1 nm/min and the length at a rate of 15 nm/min.

Write the function 𝑓 ( 𝑡 ) that gives the aspect ratio of the nanowires, the ratio of their lengths to their diameters, as a function of the growth duration 𝑡 , in minutes, after the nanowires have reached the critical size.

  • A 𝑓 ( 𝑡 ) = 2 5 0 1 5 𝑡 5 0 𝑡
  • B 𝑓 ( 𝑡 ) = 5 0 + 1 5 𝑡 2 5 0 + 𝑡
  • C 𝑓 ( 𝑡 ) = 5 0 1 5 𝑡 2 5 0 𝑡
  • D 𝑓 ( 𝑡 ) = 5 0 + 𝑡 2 5 0 + 1 5 𝑡
  • E 𝑓 ( 𝑡 ) = 2 5 0 + 1 5 𝑡 5 0 + 𝑡

The scientists want to get nanowires with an aspect ratio of 10. Use the graph to find the corresponding growth duration after the nanowires have reached the critical size.

Assuming the growth mechanism remains the same, what would the aspect ratio of the nanowires be after a very long growing time?

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