Problem solving solutions

The solutions for the questions in the previous step are below. Use these to check your method and your answers.

Question 1

A ball is thrown straight up, from 2 m above the ground, with a velocity of 15 m/s such that the height, (h) in metres at a given time (t) seconds can be represented as

[h= 2 +15t -5t^{2}]

a) By drawing a suitable graph estimate when the height is equal to 10 metres.

We first create a table of values that shows the values of (h) for various values of (t).

For example when (t=0), (h=2+15times 0 – 5 times 0^{2} = 2 m).

t	0	0.5	1	1.5	2	2.5	3	3.5
h	2	8.25	12	13.25	12	8.25	2	-6.75

We then plot these values of (h) and (t) on a graph as follows

We then draw a smooth curve through these points as follows

We can then use the graph to find the values of (t) when (h=10) as follows

Approximately when (h=10) (m), (t=0.75) (s) and (t=2.3) (s)

The question asks us to solve this question algebraically. Therefore we first write an equation for when the ball hits the ground. This is when (h=0) and so our equation is

[2+15t-5t^{2}=0]

To solve for (t) we use the quadratic equation which tells us that

[t=frac{-bpm sqrt{b^{2}-4ac}}{2a}]

In our equation (a=-5), (b=15) and (c=2), so that

[begin{eqnarray*} t=frac{-15pm sqrt{15^{2}-4times-5times 2}}{2times-5} =frac{-15pm sqrt{265}}{-10} = 3.13 text{ or } -0.128 end{eqnarray*}]

Therefore our solution is that the ball hits the ground at 3.13 seconds.

We use differentiation to find the maximum height by solving

(frac{dh}{dt}=0).

We find that (frac{dh}{dt}=15-10t) and therefore we need to solve

[begin{eqnarray*} 15-10t=0 15=10t t=1.5 end{eqnarray*}]

Then a stationary point is when (t=1.5) (s).

When (t=1.5), the height is

(h=2+15times 1.5 – 5times 1.5^{2}= 13.25) (m)

To confirm that this is a maximum height, we find the second derivative of (h) which is

[frac{d^{2}h}{dt^{2}}=-10]

This is less than 0 and therefore ((t,h)=(1.5,13.25)) is a maximum.

Question 2

We use the laws of logarithm to show that

[begin{eqnarray*} y=Ax^{k} ln(y)=ln(Ax^{k}) ln(y)=ln(A)+ln(x^{k}) ln(y)=ln(A)+kln(x) end{eqnarray*}]

such that (y’=mx’+c) where (y’=ln(y)), (x’=ln(x)), (m=k) and (c=ln(A)).

Then we find the values of (ln(y)) and (ln(x)) as in the following table

x	2	4	6
ln(x)	0.69	1.39	1.79
y	40	320	1080
ln(y)	3.69	5.77	6.98

and we plot these values on a graph as follows

We know that the relationship between (ln(y)) and (ln(x)) is (ln(y)=kln(x)+ln(A)), such that (k) is the gradient of the line and (ln(A)) is the y-intercept on the graph

We first find the gradient as

[k=m=frac{ln(y_{2})-ln(y_{1})}{ln(x_{2})-ln(x_{1})}=frac{6.98-3.69}{1.79-0.69}=3.0]

Using the graph we then find that the y-intercept is approximately 1.4 such that

(c=ln(A)=1.4).

Then we can find that

(A=e^{1.4}=4.1).

Therefore we have approximated that the relationship between (x) and (y) is (y=4.1x^{3})

For comparison the actual relationship is (y=5x^{3}).

From part (a) we know that the relationship that connects (x) and (y) will be of the form (ln(y)=kln(x)+ln(A)).

The question tells us that the lines in experiment 1 and experiment 2 are parallel and therefore the gradients are equal. Therefore the value of (k) is 3. then the relationship for experiment 2 is

[ln(y)=3ln(x)+ln(A)]

By substituting the values of (x) and (y) that are given in the question into the formula we can find the value of (A)

[begin{eqnarray*} ln(1280)=3ln(4)+ln(A) ln(A)=ln(1280)-3ln(4) ln(A)=2.9957 A=e^{2.9957…} A=20 end{eqnarray*}]

Thus the relationship between (x) and (y) which has the form (y=Ax^{k}) can be found as (y=20x^{3}).

The relationship that connects (x) and (y) is of the form (ln(y)=kln(x)+ln(A)).

The gradients of perpendicular lines, which we denote (m_{1}) and (m_{2}) are such that (m_{1}times m_{2}=-1). The gradient of the first line is 3 and therefore

[begin{eqnarray*} m_{1}times m_{2}=-1 3times m_{2}=-1 m_{2}=frac{-1}{3} end{eqnarray*}]

Therefore the value of (k) for experiment 3 is (frac{-1}{3}).

The relationship between (x) and (y) is of the form (y=Ax^{k}) and we know that (k=frac{-1}{3}), (x=1) and (y=1) so that

[begin{eqnarray*} 1=Atimes 1^{-1/3} 1=Atimes 1 A=1 end{eqnarray*}]

The relationship between (x) and (y) is (y=x^{-1/3})

Question 3

Let’s first annotate the diagram with the angles and label the opposite and adjacent in the right angled triangles.

The adjacent is equal to 25 metres in both triangles and we know that Opposite = (tan(theta) times) Adjacent. Then in the smaller triangle

[text{Opposite}=tan(61)times 25=45.1hspace{0.1cm}m]

Similarly for the larger triangle

[text{Opposite}=tan(67)times 25=58.9hspace{0.1cm}m]

Thus mast section A is 45.1 m and mast section B can be found as (58.9-45.1=13.8) m, as in the following diagram

The mast does not meet the regulations for section B as it exceeds the maximum height of 10 m.

Using the information in the question we have a triangle as follows and we need to find the length of side (a).

We can use the cosine rule so that

[begin{eqnarray*} a^{2}=b^{2}+c^{2}-2bccos(A) a^{2}=103.8^{2}+103.8^{2}-2times 103.8 times 103.8 times cos(109.3) a^{2}=28671.09499 a=sqrt{28671.09499} a=169.3hspace{0.1cm}text{pm} end{eqnarray*}]

Question 4

We first make a table of suitable values for (t) and (y). As an example when (t=0), (y(0)=4sin(3pitimes 0 -1)=-3.4)

t	y
0	-3.4
0.1	-0.2
0.2	3.1
0.3	3.9
0.4	1.5
0.5	-2.2
0.6	-4.0
0.7	-2.5
0.8	1.0
0.9	3.7
1.0	3.4

We then plot the values of (t) and (y) and join the points with a smooth curve as in the graph below.

By annotating the graph we can estimate the amplitude and period.

The amplitude is approximately 4 cm.

The period is approximately 0.95-0.28=0.67.

The frequency is (1divtext{ period}) and (1div 0.67=1.5).

Question 5

When (20leq x leq 22) we see that (y>0) and thus

[begin{eqnarray*} int_{20}^{22} 10e^{0.04x}dx = left[frac{10e^{0.04x}}{0.04}right]_{20}^{22} =left(frac{10e^{0.04times 22}}{0.04}right)-left(frac{10e^{0.04times 20}}{0.04}right) =46.34 end{eqnarray*}]

For the number of births between day 5 and day 15 we calculate

[begin{eqnarray*} int_{5}^{15}10e^{0.04x}dx=left[frac{10e^{0.04x}}{0.04}right]_{15}^{5} =frac{10}{0.04}left(e^{0.04times 15}-e^{0.04times 5}right) =150.18 end{eqnarray*}]

Similarly for the number of deaths we need to calculate (int_{5}^{15}0.1xe^{0.2x}dx)

We use integration by parts. The formula for integration by parts is

[inthspace{0.1cm}uhspace{0.1cm}frac{dv}{dx}hspace{0.1cm}dx = uv – int vhspace{0.1cm}frac{du}{dx}hspace{0.1cm}dx]

In our question (u=0.1x) and (frac{dv}{dx}=e^{0.2x}) so that (frac{du}{dx}=0.1) and (v=frac{e^{0.2x}}{0.2}). Then

[begin{eqnarray*} int_{5}^{15}0.1xe^{0.2x}dx=left[0.1xtimes frac{e^{0.2x}}{0.2}right]_{5}^{15}-int_{5}^{15}frac{0.1}{0.2}e^{0.2x}dx =left(0.1times 15 times frac{e^{0.2times 15}}{0.2}right)-left(0.1times 5 times frac{e^{0.2times 5}}{0.2}right)-left[frac{0.1}{0.2^{2}}e^{0.2x}right]_{5}^{15} =150.64-6.80-left(frac{0.1}{0.2^{2}}e^{0.2times 15}-frac{0.1}{0.2^{2}}e^{0.2times 5}right) =100.42 end{eqnarray*}]

The number of births are 150 and the number of deaths are 100 and therefore the overall population growth is (150-100=50)

Question 6

In this questions to find the maximum, we need to find a stationary point, and therefore we need to solve (frac{dy}{dt}=0).

(y=frac{7t-5}{e^{10t}}) and therefore we can differentiate this by either using the Quotient rule or by using the Product rule on (y=(7t-5)e^{-10t})

We will now use the product rule on (y=(7t-5)e^{-10t})

The Product rule is that if (y=utimes v) then (frac{dy}{dt}=ufrac{dv}{dt}+vfrac{du}{dt})

In this question (u=7t-5) and (v=e^{-10t}) so that (frac{du}{dt}=7) and (frac{dv}{dt}=-10e^{-10t}).

Then

[begin{eqnarray*} frac{dy}{dt}=(7t-5)(-10e^{-10t})+(e^{-10t})times 7 =e^{-10t}(-10(7t-5)+7) =e^{-10t}(-70t+50+7) =e^{-10t}(57-70t) end{eqnarray*}]

Then we need to find the stationary point by solving (frac{dy}{dt}=0)

(e^{-10t}(57-70t)=0) can be solved by finding when (57-70t=0)

The solution is when (t=frac{57}{70})

When (t=frac{57}{70}), is the value of (y) a maximum or a minimum? We now must find the second derivative of (y) to answer this question.

To find (frac{d^{2}y}{dt^{2}}=frac{d}{dt}left(frac{dy}{dt}right)) we use the product rule with

(u=e^{-10t}) and (v=57-70t)

so that

(frac{du}{dt}=-10e^{-10t}) and (frac{dv}{dt}=-70)

Then

[begin{eqnarray*} frac{d^{2}y}{dt^{2}}=e^{-10t}times -70 + (57-70t)times (-10e^{-10t}) =e^{-10t}(-70+10(57-70t)) =e^{-10t}(-70-570+700t) =e^{-10t}(700t-640) end{eqnarray*}]

When (t=frac{57}{70})

[frac{d^{2}y}{dt^{2}}=e^{-10times frac{57}{70}}left(700times frac{57}{70}-640right)=-0.02]

This is less than 0 and therefore (t=frac{57}{70}=0.81^{o}C) is a maximum. The maximum yield is

[y=frac{7t-5}{e^{10t}}=frac{7times frac{57}{70}-5}{e^{10times 57/70}}=2times 10^{-4}.]

Discuss in the comments below:

How did you find solving these applied problems? Why do you think it is important to practise these kinds of problems as well as those which are presented as ‘straightforward maths questions’?