Systems of Linear Equations

[Maximum mark: 6]

The Burns, Gordons and Longstaff families make meal plans for their households. The table below shows the amount of carbohydrate, fat and protein, all measured in grams, consumed by the family over a single day. The table also shows the daily calories, measured in kcal, this equates to.

Let $x$, $y$ and $z$ represent the amount of calories, in kcal, for $1$ g of carbohydrate, fat and protein respectively.

1. Write down a system of three linear equations in terms of $x$, $y$ and $z$ that represents the information in the table above. [2]
   

2. Find the values $x$, $y$ and $z$. [2]
   

The Howe family also plans meals. The table below shows the amount of carbohydrates, fat and protein consumed by the family over a single day.

3. Calculate the daily calories for the Howe family. [2]
   

[Maximum mark: 6]

A toy rocket is fired, from a platform, vertically into the air, its height above the ground after $t$ seconds is given by $s(t) = at^2 + bt + c$, where $a,b,c \in \mathbb{R}$ and $s(t)$ is measured in metres.

After $2$ second, the rocket is $28.3$ m above the ground; after $4$ seconds, $25.6$ m; after $5$ seconds, $14.7$ m.

1. 1. Write down a system of three linear equations in terms of $a$, $b$ and $c$.

   2. Hence find the values of $a$, $b$ and $c$. [3]
      

2. Find the height, above the ground, of the platform. [1]
   

3. Find the time it takes for the rocket to hit the ground. [2]
   

[Maximum mark: 6]

An owl takes off from from a tree branch and flies higher into the sky. Its height above the ground after $t$ seconds, where $t\geq 0$, is given by $s(t) = at^3 + bt^2 + ct+d$, where $a,b,c,d \in \mathbb{R}$ and $s(t)$ is measured in metres.

Initially the owl is $12\,$ metres above the ground.

1. Write down the value of $d$. [1]
   

After $1$ second, the owl is $19.8$ m above the ground; after $2$ seconds, $34.5$ m; after $4$ seconds, $22.8$ m.

2. 1. Write down a system of three linear equations in terms of $a$, $b$ and $c$.

   2. Hence find the values of $a$, $b$ and $c$. [3]
      

After some time the owl reaches a maximum height. At this time it spots some prey at ground level and then immediately swoops down to catch it.

3. 1. Find the maximum height of the owl above the ground as it spots the prey.

   2. Find the time it catches the prey. [2]
      

[Maximum mark: 8]

The graph below shows the amount of money $M$ (in thousands of dollars), in the account of a contractor, where $t$ is the time in months since he opened the account.

1. Write down one characteristic of the graph which suggests that a cubic function might be an appropriate model for the amount of money in the account. [1]
   

The equation of the model can be expressed as $M(t)=at^3+bt^2+ct+d$, where $a$, $b$, $c$ and $d \in \mathbb{R}$. It is given that the graph of the model passes through the following points.

2. 1. State the value of $d$.

   2. Using the values in the table, write down three equations in $a$, $b$, and $c$.

   3. By solving the system of equations from part (ii), find the values of $a$, $b$ and $c$. [4]
      

If $M$ has a negative value, the contractor is in debt.

3. Use the model from part (b) to find the number of months the contractor expects to be in debt. Give your answer to the nearest month. [3]
   

[Maximum mark: 12]

Coral is a wildlife expert who tags flying fish and records their movement using an electronic device.

The tagging device tells her the height of a fish relative to the water level, at any given time.

She knows that the fish swim mostly in the water, but occasionally they fly (jump!) out of the water.

The height is measured in metres and the time in seconds. If the height is negative the fish is under the water, if the height is positive the fish is flying.

Coral notices one particular fish as it flies out of the water. The moment it re-enters the water the device begins tracking its height.

At $2$ seconds the fish is at a height of $-2.8\,$m; at $5$ seconds the fish is at a height of $-2\,$m and at $11$ seconds the fish is also at a height of $-2\,$m.

The height of the fish can be expressed as $H(t)=at^3+bt^2+ct+d$, where $a$, $b$, $c$ and $d \in \mathbb{R}$.

1. 1. Write down the value of $d$.

   2. Using the information given write down three equations involving $a$, $b$ and $c$.

   3. Solve the system of equations to find the values of $a$, $b$ and $c$. [4]
      

From previous research, Coral knows that if a fish is flying for more than $1$ second then a seagull will attempt to catch it.

2. Use a justification to explain why a seagull will attempt to catch this fish. [4]
   

At $t=9\,$s a seagull begins swooping down to catch the fish.

Its height is given by $b(t)=-1.5t^2+27t-118$.

3. 1. Find the height at which the bird catches the fish.

   2. Interpret the answer in the context of the problem. [4]
      

[Maximum mark: 7]

Consider the quadratic function $f(x) = ax^2+bx+c$. The graph of $y=f(x)$ is shown in the diagram below. The vertex of the graph has coordinates $\text{R}(m,-9)$.

The graph intersects the $x$-axis at two points; $\text{P}(-4,0)$ and $\text{Q}(2,0)$.

1. Find the value of $m$. [1]
   

2. Find the values of $a$, $b$, and $c$.[5]
   

3. Write down the equation of the axis of symmetry of the graph. [1]
   

[Maximum mark: 12]

The graph below shows the profit $P$ (in thousands of dollars), that business A makes, where $t$ is the time in months since January 1st.

1. Write down one characteristic of the graph which suggests that a cubic function might be an appropriate model for the business profit. [1]
   

The model can be expressed as $P(t)=at^3+bt^2+ct+d$, where $a$, $b$, $c$ and $d \in \mathbb{R}$. It is given that the graph of the model passes through the following points.

2. 1. State the value of $d$.

   2. Using the values in the table, write down three equations in $a$, $b$, and $c$.

   3. By solving the system of equations from part (ii), find the values of $a$, $b$ and $c$. [4]
      

If $P$ has a negative value, business A is losing money. The owner has decided they will not open during the corresponding time next year.

3. Use the model from part (b) to find the approximate dates during which business A will not open next year. [4]
   

Business B has a profit function given by $P(t)=-0.1t^2+1.2t$, for $0 \leq t \leq 12$.

4. Determine the total amount of time for which business B is more profitable than business A. [3]
   

[Maximum mark: 17]

The Burj Khalifa, located in Dubai, is the tallest building in the world. It has a height of $830 \text{ m}$ and has a square base that covers a floor area of $556 \text{ m} \times 556 \text{ m}$. A tourism shop located near the building sells souvenirs of the tower, which sit inside glass pyramids, as illustrated by the diagram below. The souvenir tower is an accurate scale replica of the actual tower.

The scaled model of Burj Khalifa has a base area of $20 \text{ cm} \times 20 \text{ cm}$. The height and base area dimensions of the glass pyramid are 10% larger than the model.

1. 1. Find the height of the souvenir tower, in cm, correct to the nearest mm.

   2. Find the volume of the glass pyramid, rounding your answer correct to the nearest cubic centimetre. [5]
      

The shop owner aims to maximise profits from selling the souvenirs. The more the owner orders from the manufacturer, the cheaper the souvenirs are to buy. However, if too many are ordered, profits may decrease due to surplus stock unsold.

The number of souvenirs ordered from previous years and the resulting profits are shown in the following table.

Quantity	Profit($)
$5000$	$35\,000$
$10\,000$	$95\,500$
$13\,000$	$116\,500$

The shop owner decides to fit a cubic model of the form

to model the profit, $P$, for $x$ thousand souvenirs ordered.

2. Explain why $d=0$.[1]
   

3. Construct three linear equations to solve for $a$, $b$ and $c$, and hence write down the completed function $P(x)$. [5]
   

4. Find $P'(x)$.[2]
   

5. Find, to the nearest hundred souvenirs, the optimal number of souvenirs the owner should buy to maximise profit, and the resulting profit from this number. [4]