Practice: a quadratic model of revenue and profit

Finance and economics, grade 12 (3 units) Lesson 4 of 16Practice

Practice: a quadratic model of revenue and profit

The town swimming pool has 500 members, and each pays 120 ₪ a month.
Every rise of 10 ₪ in the monthly price drives 25 members away.
$\displaystyle x$ is the number of 10 ₪ rises.

What will the revenue a month be after 2 rises?

/ – 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

On the graph, the revenue a month after $\displaystyle x$ rises.
Mark the parabola’s vertex.

\displaystyle R(x)=(120+10x)(500-25x)

Which monthly price gives the pool the greatest revenue?
How many members will there be at that price, and what will the revenue be?

Monthly price (₪)

/ – 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

Members

/ – 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

Revenue (₪)

/ – 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 museum has $\displaystyle {q}={800-40p}$ visitors a day, where $\displaystyle p$ is the entrance price in shekels.
On the graph, $\displaystyle x$ is the entrance price, and $\displaystyle y$ the revenue a day.
Mark the points where the revenue is 0, and the vertex.
Which entrance price gives the greatest revenue?
How many visitors will there be a day, and what will the revenue be?

\displaystyle R(x)=x(800-40x)

Entrance price (₪)

/ – 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

Visitors a day

/ – 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

Revenue a day (₪)

/ – 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 food truck sells 240 portions a day, each for 30 ₪.
Every rise of 3 ₪ in the price of a portion takes 12 portions a day off the sales.
$\displaystyle x$ is the number of 3 ₪ rises, and the revenue a day is $\displaystyle {R(x)}={(30+3x)(240-12x)}$ .

Solve the equation: when is the revenue 0?

What does each of the solutions say about the price or the buyers?
With the help of the solutions, find the price of a portion that gives the greatest revenue, and that revenue.

Price of a portion (₪)

/ – 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

Revenue a day (₪)

/ – 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 painting club is a small business.
Its profit a month, in shekels, is $\displaystyle {P(x)}={-5x^{2}+300x-4000}$ , where $\displaystyle x$ is the price of a lesson in shekels.

Which lesson price gives the greatest profit?

/ – 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

What is the greatest profit 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

Solve the equation: what are the break-even prices?

A concert hall sells $\displaystyle {q}={1000-10p}$ tickets for a show, where $\displaystyle p$ is the price of a ticket in shekels.
Each member of the audience costs the hall 10 ₪ (ushers, cleaning).
The fixed costs of a show are 14,000 ₪.
The profit from a show, where $\displaystyle x$ is the ticket price, is $\displaystyle {P(x)}={x(1000-10x)-(14000+10(1000-10x))}$ .

Expand the brackets and collect like terms.

/ – 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

Write an equation and solve it: at which ticket prices does the hall neither make a profit nor a loss?

Which ticket price gives the greatest profit?
How many tickets will sell at that price, and what will the profit from a show be?

Ticket price (₪)

/ – 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

Tickets

/ – 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

Profit from a show (₪)

/ – 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 juice stall sells $\displaystyle {q}={300-7p}$ cups of juice a day, where $\displaystyle p$ is the price of a cup in shekels.

Which price of a cup gives the greatest revenue?
Round to agorot (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

What is the greatest revenue a day, rounded to shekels?

/ – 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

On the graph, the weekly profit of an electric-bike shop, in shekels, where $\displaystyle x$ is the price of a day’s hire in shekels.
The formula is hidden: $\displaystyle A$ and $\displaystyle B$ are where the graph meets the $\displaystyle x$ -axis, and $\displaystyle V$ is the vertex.
Write the formula of the profit on the graph.
At which prices does the shop make a profit, and what is the greatest profit a week?

\displaystyle P(x)=\,?

\displaystyle A=(10,0)

\displaystyle B=(50,0)

\displaystyle V=(30,800)

Greatest profit (₪)

/ – 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 shopkeeper says: “The more I raise the price, the more I earn on each item, so my profit will grow.”
Is he right?
Explain with the help of the profit’s parabola.

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

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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 $\displaystyle z$
$\displaystyle x=\mu+z\cdot\sigma$