Lesson Worksheet: Numerical Methods in Real-Life Situations Mathematics

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In this worksheet, we will practice using numerical methods to solve real-life problems.

Q1:

The temperature, π‘‡βˆ˜C, in a city on a specific day can be modeled by the equation 𝑇(β„Ž)=βˆ’15,000(β„Ž+4)(β„Žβˆ’4)(β„Žβˆ’22)+3, where β„Ž is the number of hours past midnight and 0β‰€β„Žβ‰€24. Consider the temperature at the start and the end of the day. Does this tell us anything about whether the temperature reaches 0∘C?

  • AYes, it reaches 0∘C.
  • BYes, the temperature remains above 0∘C all day.
  • CNo, it does not tell us anything.
  • DYes, the temperature remains below 0∘C all day.

Q2:

The height, β„Ž meters, of a projectile above the ground after 𝑑 seconds is given by β„Ž(𝑑)=βˆ’π‘‘+2π‘‘βˆ’π‘‘+3𝑑οŠͺ, for 𝑑β‰₯0. A graph of β„Ž(𝑑) is given in the figure.

The equation β„Ž(𝑑)=0 can be rewritten in the form 𝑑=√2π‘‘βˆ’π‘‘+3π‘‘οŽ£οŠ©οŠ¨. Given that the projectile hits the ground between 2 and 3 seconds after launch, use this equation and an initial value of 𝑑=2.2seconds to find the time taken for the projectile to hit the ground correct to one decimal place.

Use this value to suggest an improvement to the range of validity of the model.

  • A0≀𝑑≀2.2
  • B0≀𝑑≀2.4
  • C2≀𝑑≀2.4
  • D0≀𝑑≀2.3
  • E0≀𝑑≀2

Q3:

A particle in motion has a height of β„Ž meters above the ground given by the equation β„Ž(𝑑)=ο€Ήπ‘‘βˆ’π‘‘+1+4𝑑lnsin, where 𝑑 is the time in seconds and 0β‰€π‘‘β‰€π‘‘οŒ¦. π‘‘οŒ¦ is the time taken for the particle to hit the ground.

Apply the Newton–Raphson method with initial value 𝑑=3 to find a second approximation of π‘‘οŒ¦. Give your answer to two decimal places.

Q4:

The mass of living competing bacteria, in milligrams, in a Petri dish can be modeled by the formula π‘š(𝑑)=𝑒(𝑑)βˆ’π‘‘+4ln, where 𝑑 is the time in hours, 𝑑β‰₯0, and π‘šβ‰₯0.

By applying the Newton–Raphson method a sufficient number of times with initial value 𝑑=2, find the time taken for the mass of bacteria to die out correct to one decimal place.

Q5:

A ball is rolling down into a sink. Its distance 𝑑 cm from the drain at time 𝑑 s can be modeled by the equation 𝑑(𝑑)=π‘‘βˆ’π‘‘+3tan, for 𝑑β‰₯0.

Given that the ball falls into the drain between 1 s and 2 s after launch, use the iterative formula 𝑑=𝑑+3ο…οŠοŠ°οŠ§οŠ±οŠ§οŠοŠ¨tan to find the time taken for the ball to fall into the drain correct to one decimal place.

Use this value to suggest an improvement for the range of validity of the model.

  • A0≀𝑑≀1.4
  • B1.3≀𝑑≀2
  • C1.4≀𝑑≀2
  • D0≀𝑑≀1.3

Q6:

A scientist models the height of a falling particle by the equation β„Ž(𝑑)=π‘’βˆ’π‘‘sin, 𝑑β‰₯0.

Take 𝑑=2 as a first approximation to the time at which the particle hits the ground and apply the Newton-Raphson method once to suggest an improvement to the range of validity of the model.

Give your answer to two decimal places.

  • A0≀𝑑≀1.61
  • B0≀𝑑≀2.61
  • C0≀𝑑≀2.22
  • D0≀𝑑≀2.10
  • E0≀𝑑≀1.82

Q7:

The height of a rollercoaster above ground level in meters follows the graph of β„Ž(π‘₯)=ο€Ήπ‘₯+1ο…βˆ’π‘₯lnsin, where π‘₯ is the distance in meters along the ground and 0≀π‘₯≀4.

Which of the following is the correct formulation of the Newton–Raphson method to approximate when the rollercoaster will return to ground level?

  • Aπ‘₯=π‘₯βˆ’ο€Ήπ‘₯+1ο…βˆ’π‘₯+π‘₯οŠοŠ°οŠ§οŠοŠ¨οŠοŠοŠ¨ο—ο—οŠ°οŠ§οŠlnsincosο‘ƒοŽ‘ο‘ƒ, π‘₯=0
  • Bπ‘₯=π‘₯+ο€Ήπ‘₯+1ο…βˆ’π‘₯βˆ’π‘₯οŠοŠ°οŠ§οŠοŠ¨οŠοŠοŠ¨ο—ο—οŠ°οŠ§οŠlnsincosο‘ƒοŽ‘ο‘ƒ, π‘₯=0
  • Cπ‘₯=π‘₯+ο€Ήπ‘₯+1ο…βˆ’π‘₯βˆ’π‘₯οŠοŠ°οŠ§οŠοŠ¨οŠοŠοŠ¨ο—ο—οŠ°οŠ§οŠlnsincosο‘ƒοŽ‘ο‘ƒ, π‘₯=2
  • Dπ‘₯=π‘₯βˆ’ο€Ήπ‘₯+1ο…βˆ’π‘₯βˆ’π‘₯οŠοŠ°οŠ§οŠοŠ¨οŠοŠοŠ¨ο—ο—οŠ°οŠ§οŠlnsincosο‘ƒοŽ‘ο‘ƒ, π‘₯=0
  • Eπ‘₯=π‘₯βˆ’ο€Ήπ‘₯+1ο…βˆ’π‘₯βˆ’π‘₯οŠοŠ°οŠ§οŠοŠ¨οŠοŠοŠ¨ο—ο—οŠ°οŠ§οŠlnsincosο‘ƒοŽ‘ο‘ƒ, π‘₯=2

Q8:

The depth of a hole β„Ž in meters is given by the formula β„Ž(𝑀)=0.5(𝑀)π‘€βˆ’3sec, where 𝑀 is the distance from the center of the hole in meters and β„Žβ‰€0.

Apply the Newton–Raphson method with initial value 𝑀=1.25 to find the next four approximations of the distance from the center of the hole to the edge. Give your answers to three decimal places.

  • A𝑀=1.296, 𝑀=1.290, 𝑀=1.290, 𝑀=1.290οŠͺ
  • B𝑀=3.154, 𝑀=3.165, 𝑀=3.176, 𝑀=3.187οŠͺ
  • C𝑀=1.554, 𝑀=1.145, 𝑀=1.135, 𝑀=1.125οŠͺ
  • D𝑀=1.290, 𝑀=1.290, 𝑀=1.290, 𝑀=1.296οŠͺ
  • E𝑀=1.125, 𝑀=1.135, 𝑀=1.145, 𝑀=1.554οŠͺ

Q9:

The population 𝑃 of a city after 𝑑 years can be modeled by the equation 𝑃(𝑑)=1100π‘’βˆ’π‘‘οŠ¦οŽ–οŠ§οŠ§οοŠ©, for 𝑑β‰₯0.

By considering the given graph of this function, determine which of the following is a problem with this model.

  • AThe model predicts that the population will stay steady for the first 25 years.
  • BThe model predicts that the population will grow with bounds.
  • CThe model predicts that the population will eventually be negative.
  • DThe model predicts that the population will eventually start to decrease.
  • EThe model predicts that the population will eventually grow too rapidly.

There is a stationary point on the curve 𝑦=𝑃(𝑑) in the interval 12<𝑑<14. Using the Newton–Raphson method with 𝑑=13, find the next approximation of when the rate of change of the population is 0. Give your answer to two decimal places.

Q10:

A given wave in an ocean has an amplitude given by the function 𝐴(𝑑)=18𝑑+1√2sin, where 𝐴 is the amplitude of the wave in metres above sea level, 𝑑 is the time in seconds, and 0≀𝑑≀3. The graph of the amplitude of the wave is given as follows.

Which of the following is the correct formulation of the Newton–Raphson method to approximate when the wave will be at sea level?

  • A𝑑=π‘‘βˆ’π‘‘+2π‘‘π‘‘οŠοŠ°οŠ§οŠοŠ¨οŠοŠ§βˆšοŠ¨οŠοŠ¨οŠsincos, 𝑑=1.8
  • B𝑑=π‘‘βˆ’π‘‘+2π‘‘π‘‘οŠοŠ°οŠ§οŠοŠ¨οŠοŠ§βˆšοŠ¨οŠοŠ¨οŠsincos, 𝑑=0
  • C𝑑=π‘‘βˆ’π‘‘2π‘‘π‘‘οŠοŠ°οŠ§οŠοŠ¨οŠοŠοŠ¨οŠsincos, 𝑑=1.25
  • D𝑑=π‘‘βˆ’π‘‘+2π‘‘π‘‘οŠοŠ°οŠ§οŠοŠ¨οŠοŠ§βˆšοŠ¨οŠοŠ¨οŠsinsin, 𝑑=1.8
  • E𝑑=π‘‘βˆ’π‘‘2π‘‘π‘‘οŠοŠ°οŠ§οŠοŠ¨οŠοŠοŠ¨οŠsinsin, 𝑑=6

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