MHF4U U04A06 Homework Solutions

  1. The water at a local beach has an average depth of 1 meter at low tide. The average depth of the water at high tide is 8 m. If one cycle takes 12 hours:
  1. Determine the equation of this periodic function using cosine as the base function where a time of 0 is the beginning of high tide. t = 0 is midnight.

Solution:

  1. What is the depth of the water at 2 a.m.

Solution: 6.25 m

  1. Many people dive into the water from the nearby dock. If the water must be at least 3 m deep to dive safely, between what daylight hours should people dive?

Solution: They should not between the hours 3:50 a.m. – 9:24 a.m. and 3:50 p.m. – 9:24 p.m.

  1. The data below represent the average monthly temperature in Ottawa.

Jan / Feb / Mar / Apr / May / June / July / Aug / Sept / Oct / Nov / Dec
0C / -6.4 / -5.6 / -0.8 / 6.3 / 12.3 / 17.6 / 20.7 / 19.7 / 15.4 / 9.1 / 3.2 / -3.3

Using Fathom, the following graph was created.

a)  Find the equation that best models this data using the sine function as the base.

b)  The average temperature could increase by 2 oC over the next 50 years due to global warming. What would the new equation to your graph be if the global temperature increase by 2 oC.

  1. A nail located on the circumference of a water wheel is moving as the current pushes on the wheel. The height of the nail in terms of time can be modeled by the graph shown below. Determine the equation of a sinusoidal function that models this situation.

Solution:

  1. The top of a flagpole sways back and forth in high winds. The top sways 10 cm to the right (+10 cm) and 10 cm to the left (-10 cm) of its resting position and moves back and forth 240 times every minute. At t = 0, the pole was momentarily at its resting position. Then it started moving to the right.
  1. Determine the equation to model this function that describes the distance the top of the pole is from its resting position in terms of time using both a both a base function of sine then using a base function of cosine.

Solution: and

  1. If the wind speed decreases slightly such that the sway of the top of the pole is reduced by 20%, what is the new equation of the sinusoidal function? Assume that the period remains the same.

Solution: losing 20% means keeping 80%, therefore 80% of 10 is 8.

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