Practice: analytic geometry

Finance and economics, grade 12 (3 units) Lesson 12 of 16Practice

Practice: analytic geometry

On the city map, a shoe shop is at the point $\displaystyle A(-3,2)$ , and a clothes shop at the point $\displaystyle B(5,8)$ .
One unit on each axis is 100 metres.

Work out the straight-line distance between the shops, in units.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A car park is built at the midpoint of the segment $\displaystyle AB$ .
Find its $\displaystyle x$ -coordinate.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Find the car park’s $\displaystyle y$ -coordinate.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A delivery drone sets out from the warehouse $\displaystyle D(1,1)$ , brings a parcel to the customer $\displaystyle E(4,5)$ , from there to the customer $\displaystyle F(10,-3)$ , and returns to the warehouse.
It flies in straight lines, and one unit on each axis is 100 metres.

Work out the distance from $\displaystyle D$ to $\displaystyle E$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Work out the distance from $\displaystyle E$ to $\displaystyle F$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Work out the distance from $\displaystyle F$ back to $\displaystyle D$ , rounded to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

About how many metres does the drone fly on the whole round?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

An electrician stretches a straight cable from a pole at the point $\displaystyle P(1,7)$ to a building at the point $\displaystyle Q(6,2)$ .
One unit on each axis is one metre.

Work out the length of the cable in metres, rounded to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The cable is sold in whole metres, at 12 ₪ a metre.
How much will the cable the electrician needs to buy cost?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A light-rail line runs in a straight line through the stops $\displaystyle (2,1)$ and $\displaystyle (6,9)$ .
Write the line’s equation on the graph board.
Is the stop $\displaystyle (5,7)$ on the line? Substitute and explain.

\displaystyle (2,1)

\displaystyle (6,9)

A straight fence passes through the points $\displaystyle A$ and $\displaystyle B$ on the graph board.
Write the fence’s equation on the graph board.

\displaystyle y=\,?

\displaystyle A=(-2,5)

\displaystyle B=(4,2)

The town’s main road is $\displaystyle {y}={-3x+5}$ .
Write on the graph board the equation of a road parallel to it, passing through the petrol station $\displaystyle P(2,1)$ .

\displaystyle y=-3x+5

\displaystyle P=(2,1)

Two roads: $\displaystyle {y}={0.5x+1}$ , and $\displaystyle {y}={-x+7}$ (the dashed one).
Mark their junction on the graph, and solve the system of equations to find its exact coordinates.

\displaystyle y=0.5x+1

\displaystyle y=-x+7

$\displaystyle x$ is the number of minutes of calls a month, and $\displaystyle y$ the price in shekels.
The graph of plan A is a line through $\displaystyle (0,20)$ and $\displaystyle (100,50)$ .
The graph of plan B is a line through $\displaystyle (0,40)$ and $\displaystyle (100,60)$ .

Write the equations of the two plans on the graph board.

\displaystyle (0,20)

\displaystyle (100,50)

\displaystyle (0,40)

\displaystyle (100,60)

Write the system of equations and solve it: for how many minutes do the two plans cost the same, and what is that price?

Noa talks for about 300 minutes a month.
Which plan is better for her, and how much will she save a month?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The warehouse $\displaystyle M(4,6)$ is at the midpoint of the segment between the shop $\displaystyle A(1,2)$ and the shop $\displaystyle B$ .
Find the coordinates of $\displaystyle B$ .

$\displaystyle x$ is the $\displaystyle x$ -coordinate of $\displaystyle B$ :

$\displaystyle y$ is the $\displaystyle y$ -coordinate of $\displaystyle B$ :

Formulas: finance and economics, grade 12 (3 units)

Open as a page

Revenue and profit: a quadratic model

The revenue – $\displaystyle p$ the price of a unit, $\displaystyle q$ the number of units sold
$\displaystyle R=p\cdot q$
The profit – the revenue minus the cost
$\displaystyle P=R-C$
A parabola – the revenue or the profit as a function of the price
$\displaystyle y=ax^{2}+bx+c$
The vertex’s $\displaystyle x$ – where the revenue or the profit is greatest (when $\displaystyle a<0$ )
$\displaystyle x=-\frac{b}{2a}$ $\displaystyle a\neq0$
The quadratic formula – the prices where the revenue or the profit is 0
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

Linear programming

A constraint – an inequality; the points that satisfy it are a half-plane
$\displaystyle ax+by\le c$
The objective function – the greatest profit is at one of the feasible region’s corner points
$\displaystyle P=ax+by$

Analytic geometry

The distance between the points $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$
$\displaystyle d=\sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}$
The midpoint of the segment between the two points
$\displaystyle M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$
The slope of the line through the two points
$\displaystyle m=\frac{y_2-y_1}{x_2-x_1}$
The equation of the line through the point $\displaystyle (x_1,y_1)$ with slope $\displaystyle m$
$\displaystyle y-y_1=m(x-x_1)$
Parallel lines – the same slope
$\displaystyle m_1$ $\displaystyle =$ $\displaystyle m_2$

The normal distribution

In a normal distribution – $\displaystyle \mu$ the mean, $\displaystyle \sigma$ the standard deviation: about 68% of the data within one standard deviation of the mean (34% on each side)
$\displaystyle P(\mu-\sigma<x<\mu+\sigma)\approx68\%$
About 95% within two standard deviations of the mean (another 13.5% on each side)
$\displaystyle P(\mu-2\sigma<x<\mu+2\sigma)\approx95\%$
About 2.5% above $\displaystyle \mu+2\sigma$ , and about 2.5% below $\displaystyle \mu-2\sigma$
$\displaystyle P(x>\mu+2\sigma)\approx2.5\%$
The standard score of the value $\displaystyle x$ – how many standard deviations it is above the mean (or below it, when $\displaystyle z<0$ )
$\displaystyle z=\frac{x-\mu}{\sigma}$
The value whose standard score is $\displaystyle z$
$\displaystyle x=\mu+z\cdot\sigma$