Science:Math Exam Resources/Courses/MATH110/December 2015/Question 10 (a)

MATH110 December 2015

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[hide] Question 10 (a)

Upon the first snow fall of the year, your friends in Ontario always place a metre stick in their yard so they can measure how much snow they get throughout the winter. Their measurements for last year’s snow during the month of December are shown in the table below. Day in December 1 9 11 14 18 24 31 Amount of snow (cm) 19 8 23 14 34 29 33 (a) According to the table, what is the minimum number of times your friends must have had exactly 20 cm of snow in their yard? Justify your answer.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

[show] Hint
Use the Intermediate Value Theorem.

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[show] Solution

Let's first recall the statement of the Intermediate Value Theorem: If $f$ is a continuous function on the interval $[a,b]$ and $M$ is a number between $f(a)$ and $f(b)$, then there must be a number $c$ in the interval $(a,b)$ such that $f(c)=M$ . Now we apply IVT in the context of the given problem. First, we note that since the metre stick is always placed in the yard, so it is continuously measuring the height of snow, this means that the function of height, $f$, is a continuous function in the interval of the days in December i.e. $[1,31]$ . The target value $20$ cm is between $f(1)=19$ cm and $f(31)=33$ cm, so we know that at least there is one day say $c$th day such that the amount of snow has been $20$ cm. However, if we look closely to the given table we see that we can split the interval $[1,31]$ into smaller subintervals $[a,b]$ such that for each subinterval, the number $20$ is between $f(a)$ and $f(b)$: $f(9)=8,\ f(11)=23\ {\text{so}}\ 8<20<23\Rightarrow$ there is a $c$th day between 9th and 11th day such that $f(c)=20$ $f(11)=23,f(14)=14\ {\text{so}}\ 14<20<23\Rightarrow$ there is a $c$th day between 11th and 14th day such that $f(c)=20$ $f(14)=14,\ f(18)=34\ {\text{so}}\ 14<20<34\Rightarrow$ there is a $c$th day between 14th and 18th day such that $f(c)=20$ Therefore, there are at least $\color {blue}3$ days that there has been exactly 20 cm of snow.