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The price for the greatest revenue

Finance and economics, grade 12 (3 units) Lesson 2 of 16Learning

The price for the greatest revenue

The greatest revenue: the vertex
Check: 150⋅300=45000\displaystyle {150\cdot300}={45000}.

The gym’s revenue after x\displaystyle x rises of 5 ₪ is R(x)=−50x2+1000x+40000\displaystyle {R(x)}={-50x^{2}+1000x+40000}.
The parabola opens downwards, so its highest point is the vertex – where the revenue is greatest.

The x\displaystyle x-coordinate of the vertex of y=ax2+bx+c\displaystyle {y}={ax^{2}+bx+c}:

x\displaystyle x =\displaystyle = −b2a\displaystyle -\frac{b}{2a}a≠0\displaystyle a\neq0

The example below finds the best price:

  • a=−50\displaystyle {a}={-50}, b=1000\displaystyle {b}={1000}, so x=−10002⋅(−50)=10\displaystyle {x}={-\frac{1000}{2\cdot(-50)}}={10} rises.
  • The price: 100+5⋅10=150\displaystyle {100+5\cdot10}={150} ₪, and the number of members: 400−10⋅10=300\displaystyle {400-10\cdot10}={300}.
  • The greatest revenue: R(10)=−50⋅100+1000⋅10+40000=45000\displaystyle {R(10)}={-50\cdot100+1000\cdot10+40000}={45000} ₪.
R(x)=−50x2+1000x+40000\displaystyle R(x)=-50x^{2}+1000x+40000
V=(10,45000)\displaystyle V=(10,45000)
The prices at which the revenue is 0

The revenue is 0 when one of the factors of the product is 0 – the price or the number of customers:

(100+5x)(400−10x)=0(100+5x)(400-10x)=0
  • 100+5x=0\displaystyle {100+5x}={0}, that is x=−20\displaystyle {x}={-20}: the price is 0 ₪.
  • 400−10x=0\displaystyle {400-10x}={0}, that is x=40\displaystyle {x}={40}: the price is 300 ₪, and nobody stays a member.

These are the points where the parabola meets the x\displaystyle x-axis. The parabola is symmetric, so the vertex is halfway between them:

xV=x1+x22\displaystyle x_V=\frac{x_1+x_2}{2}

Here: xV=−20+402=10\displaystyle {x_V}={\frac{-20+40}{2}}={10} – the same vertex.

A quantity that falls in a straight line

When q=k−mp\displaystyle {q}={k-mp} units sell at a price p\displaystyle p, the revenue is R(p)=p(k−mp)\displaystyle {R(p)}={p(k-mp)}.

  • The revenue is 0 at p=0\displaystyle {p}={0}, and at the price at which nobody buys, p=km\displaystyle {p}={\frac{k}{m}}.
  • The best price is halfway between them – half the price at which nobody buys:
p=k2m\displaystyle p=\frac{k}{2m}

At the bakery of the last lesson, q=500−20p\displaystyle {q}={500-20p}: nobody buys at 25 ₪, so the best price is 12.5 ₪.
At that price 500−20⋅12.5=250\displaystyle {500-20\cdot12.5}={250} loaves sell, and the revenue is 12.5⋅250=3125\displaystyle {12.5\cdot250}={3125} ₪.

Exercise 1

At the cinema of the last lesson, the revenue from a screening after x\displaystyle x rises of 2 ₪ is R(x)=(40+2x)(300−10x)\displaystyle {R(x)}={(40+2x)(300-10x)}.
With the brackets expanded: R(x)=−20x2+200x+12000\displaystyle {R(x)}={-20x^{2}+200x+12000}.

a

Mark the parabola’s vertex on the graph.

R(x)=−20x2+200x+12000\displaystyle R(x)=-20x^{2}+200x+12000

b

Work out the number of rises that gives the greatest revenue.

    / – 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

    c

    What ticket price gives the greatest revenue, and what is that revenue?

    Ticket price (₪)

      / – 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

      Revenue (₪)

        / – 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

        Exercise 2

        The revenue of the cinema of Exercise 1 is 0 when the price is 0, or when nobody buys a ticket.

        a

        Solve the equation.

          b

          What is the ticket price at each of the solutions?
          Show that the vertex you found in Exercise 1 is halfway between the solutions.

          Smaller solution (₪)

            / – 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

            Larger solution (₪)

              / – 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

              Exercise 3

              A beach kiosk sells q=600−30p\displaystyle {q}={600-30p} ice lollies a day, where p\displaystyle p is the price of an ice lolly in shekels.
              On the graph, x\displaystyle x is the price of an ice lolly, and y\displaystyle y the revenue a day.

              a

              Mark on the graph the points where the revenue is 0, and the vertex.

              R(x)=x(600−30x)\displaystyle R(x)=x(600-30x)

              b

              At what price should an ice lolly be sold?
              How many ice lollies will sell a day at that price, and what will the revenue be?

              Price of an ice lolly (₪)

                / – 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

                Ice lollies a day

                  / – 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

                  Revenue a day (₪)

                    / – 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

                    Exercise 4

                    A station on the Yarkon river hires out 60 kayaks a day, each for 80 ₪.
                    Every rise of 10 ₪ in the price takes 5 kayaks off the hires.
                    x\displaystyle x is the number of 10 ₪ rises.

                    a

                    The revenue a day is R(x)=(80+10x)(60−5x)\displaystyle {R(x)}={(80+10x)(60-5x)}.
                    Expand the brackets.

                      / – 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

                      b

                      Work out the number of rises that gives the greatest revenue.

                        / – 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

                        c

                        At what price should a kayak be hired out, how many kayaks will be hired, and what will the revenue a day be?

                        Price of a kayak (₪)

                          / – 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

                          Kayaks a day

                            / – 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

                            Revenue a day (₪)

                              / – 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