A government has a piecewise function, T, to calculxwzehmv7 0hk,z wx f5 owr;wg4)u3n(;ate the amount of taxes a citizen h5g ; x4zvw)u7w;hx (,m0whf3wo ze nrkas to pay quarterly, according to their quarterly income, x, given by



1.Find the quarterly taxes a citizen has to pay for incomes of
(1)9000 dollars   
(2)16000 dollars   
The government is considering increasing the tax rates and is contemplating the following two options:
Option 1: Increase taxes by 100 dollars on all levels, and then by a factor of 1.15
Option 2: Increase taxes using the model T(1.4x+1000)
2.Determine functions to model the taxes for each of these options.

A soccer ball is kicked from the ground such that its path can be modelled by tn+ /zxa b-/n8a (drt;5xmglrch/(bn; c mxgardnzt-lr5 8ax+ /e function
$h(d)=d−\frac{d^2}{6}​​$
where ℎh is the height above ground level in meters, and d is the horizontal distance the ball travels, also in meters.
1.Find the distance travelled by the ball when it hits the ground.
2.Find the maximum height the ball reaches.
The soccer player now practices kicking the ball shorter, so that it travels half the vertical height and half the horizontal distance as the previous kick.
3.Determine the height function, ℎ2h2​, that describes this new trajectory.

  Let f(x)=sinx, for x∈R. The graph of f is translat4*d e3 8xicluwcl-;e ged by $ \begin{Bmatrix} p \\ -q \end{Bmatrix} $ to give the graph of a function g that passes through the points A(0,−7) and B(π,−5). It is given that 0Find the value of p and the value of q. q =   

  Let f(x)=$e^x$, for x∈R. The graph of f is translated by $ \begin{Bmatrix} -p \\ q \end{Bmatrix} $​ to give the graph of a function g that passes through the points A(0,1.5) and B(ln2,2).
Find the value of p and the value of q. q =   

A scientist is studying the growth of a population of bacteria under dceaiw0 txq ),8ifferent environments.)8 e ci,w0xtqa In environment A bacteria grows according to the function
$G(t)=−700(0.95)^t$+750
where G is the population of bacteria and t is the time in minutes since placed in the environment.
1.Find the initial population of bacteria.
2.Sketch a graph of G against t for 0≤t≤800.
3.Find the population of bacteria after 20 minutes.
4.Find the steady-state population of bacteria after a long period of time.
To slow down the growth of bacteria, the scientist creates a new environment, B, by reducing the pH level of the environment. The scientist finds that the bacteria now grow half as fast as in environment A.
5.Determine a model for the growth of bacteria $G_2$​ for this modified environment.
The scientist creates a another environment, C, which is rich in proteins. The scientist finds that the bacteria grows twice as fast as in the original environment, A.
6.Determine a model for the growth, $G_3$​, of bacteria in this protein-rich environment.
7.Find out how long it takes for the population of bacteria to reach 600 in environment C.

A company that produces electronic co1y5gzd 3yn np3i;m7nkmponents is investigating the heating properties of a particular component. The company finds thag5zk3n7i y3dn; ynp1m t the temperature, T, of the component can be modelled by the function
T(t)=$−40(0.9)^t$+90
where T is measured in degrees Celsius and t is measured in minutes of use.
1.Write down the initial temperature of the component.
2.Sketch a graph of T against t for 0≤t≤90.
3.Find the temperature of the component after 15 minutes.
4.Find the steady-state temperature of the component after a long duration of use.
To slow down the heating speed of the component, the company increases the size of the component in such a way that the component heats $\frac{3}{4}$ times as fast as the initial component.
5.Find a model for the temperature $T_2$​ of the modified component.
Another option to reduce the rate of heating - instead of increasing the size of the component - is to add a thermal pad to the component. In this case the component heats half as fast as the initial component.
6.Find a model for the temperature $T_3$​ of the component with the thermal pad.
7.Find the time taken for the component with the thermal pad to reach $80^{\circ}C$.