SL AI Paper 2 — long-answer practice

An industrial machine costs $\mathrm{\$}80000$ new. Its value depreciates by $18\mathrm{\%}$ each year. The maintenance cost is $\mathrm{\$}800$ in the first year, and increases by $\mathrm{\$}350$ each subsequent year.

1. Calculate the value of the machine at the end of year 8.

2. Calculate the maintenance cost during year 8.

3. Calculate the total cumulative maintenance cost over the first 8 years.

4. The annual depreciation loss in year $n$ is defined as the value at the start of the year minus the value at the end of the year. Show that the depreciation loss in year $n$ is given by $D_n=14400(0.82{)}^{n-1}$.

5. The machine should be replaced when the annual maintenance cost exceeds the annual depreciation loss. Using your GDC, find the year $n$ in which the machine should be replaced.

A woman saves for her retirement over a 60-year period, divided into two phases.
Phase 1 (Saving): She deposits $\mathrm{\$}1200$ at the end of every month for 35 years into an account paying $6\mathrm{\%}$ nominal annual interest, compounded monthly.

1. Find the total amount in her account at the end of the 35 years.

Phase 2 (Retirement): She transfers the total amount into a secure fund paying $4.2\mathrm{\%}$ nominal annual interest, compounded monthly. She withdraws a fixed amount, $W$, at the end of each month for 25 years, until the account balance is exactly zero.

2. Calculate the monthly withdrawal amount, $W$.

3. Calculate the total amount of money she withdrew over the 25 years.

4. Calculate the overall total interest she earned across the entire 60-year period.

The value of Car A (in thousands of dollars) is modelled by an exponential decay function $V_A(t)=20(0.8{)}^t$. The value of an Antique Car B (in thousands of dollars) is modelled by an exponential growth function $V_B(t)=10(1.05{)}^t$. The graphs are shown below.

1. Determine which graph (the solid blue line or the dashed red line) represents Car A.

2. Using your GDC, find the exact time $t$ when both cars are worth the exact same amount, and state this value.

A farmer uses 40 metres of fencing to form a rectangular enclosure against a straight wall (no fence needed along the wall). The width is $x$ metres. The area is given by $A(x)=40x-2x^2$. The graph is shown below.

1. State the contextual domain for $x$ (the possible widths of the enclosure).

2. Using the graph or otherwise, find the maximum area and state the contextual range of $A(x)$.

Two schools are represented by points $A(2,20)$ and $B(14,24)$ on a map. A road, represented by the line $R$ with equation $-x+y=4$, passes near the schools. A town planner is asked to determine the location of a new bus stop on the road such that it is exactly the same distance from the two schools.

1. Find the equation of the perpendicular bisector of [AB]. Give your equation in the form $y=mx+c$.

2. Determine the exact coordinates of the point on road $R$ where the bus stop should be located.

The equation of a straight coastline is modelled by the line $L_1:2y-x-10=0$. A boat is anchored at point $M(8,18)$. The coastguard needs to find the shortest distance from the boat to the coastline.

1. Find the gradient of the coastline $L_1$.

2. Find the equation of the line $L_2$, which passes through $M$ and is perpendicular to $L_1$.

3. Find the coordinates of point $D$, the intersection of $L_1$ and $L_2$.

4. Calculate the shortest distance from the boat to the coastline.

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A factory manufacturing computer chips has five conveyor belts. An engineer suspects that the proportion of defective chips is not evenly distributed across the five belts. On a given day, he records the following defectives:

He conducts a ${\chi }^2$ goodness-of-fit test at a $1\mathrm{\%}$ level of significance.

1. Calculate the total number of defectives recorded.

2. Estimate the expected number of defectives from each conveyor belt if the defects are evenly distributed.

3. Use your graphic display calculator to find the $p$-value.

4. State the conclusion of the test, justifying your answer.

An engineer models the stress on a bridge cable using the function $S(x)=\frac{50x}{x^2+4}$, where $x$ is the distance in metres from the center pillar, and $S$ is measured in megapascals (MPa).

1. Evaluate the limit of $S(x)$ as $x\to \mathrm{\infty }$. Explain what this implies about the stress on the cable far away from the pillar.

2. Find the average rate of change of stress between $x=0$ and $x=2$.

3. Use your graphic display calculator to find the instantaneous rate of change of stress at $x=2$.

4. Graph the function $S(x)$ on your calculator. Use the ‘G-Solv‘ maximum feature to find the distance $x$ where the stress is greatest.

5. Verify that at the maximum point found in part (d), the instantaneous rate of change $S^{\prime }(x)$ is equal to zero.

Let $f(x)=p{x}^3+q{x}^2-4x+1$, where $p$ and $q$ are constants. The graph of $f(x)$ has a local minimum point at $x=2$, meaning $f^{\prime }(2)=0$. The graph also has a point of inflection at $x=1$, meaning $f^{″}(1)=0$.

1. Find $f^{\prime }(x)$ in terms of $p$, $q$, and $x$.

2. Find $f^{″}(x)$ in terms of $p$, $q$, and $x$.

3. Set up a system of two linear equations using the given conditions at $x=2$ and $x=1$.

4. Solve the system to find the values of $p$ and $q$.