Chapter 9.1, Problem 64PPS

(a)

To determine

To Write:

The function in standard form.

Answer to Problem 64PPS

The function in standard form is :

  $R=-10p^2+100p+750$

Explanation of Solution

Given Information:

The marching band is selling poinsettias to buy new uniforms. Last year the band charged $5 each, and they sold 150. They want to increase the price this year, and they expect to lose 10 sales for each $1 increase. The sales revenue R, in dollars, generated by selling the poinsettias is predicted by the function $R=(5+p)(150-10p)$ , where p is the number of $1 price increases.

Concept Used :

• Quadratic Functions:
• Quadratic functions are non-linear and are of the form:

  $f(x)=a{x}^2+bx+c;a\ne 0$

This is the standard form of Quadratic function.

• Shape of the graph of a Quadratic function is called Parabola.
• Parabolas are symmetric about a central line called Axis Of Symmetry.
• The axis of symmetry intersects a parabola only at one point , called the Vertex.

Calculation:

  $\begin{array}{l}R=(5+p)(150-10p)\mathrm{..............................}[given\_equation]\\ R=5(150-10p)+p(150-10p)\mathrm{..............}[distributive\_property\_of\_multiplication\_over\_addition]\\ R=5(150)-5(10p)+p(150)-p(10p)\mathrm{.....}[distributive\_property\_of\_multiplication\_over\_subtraction]\\ R=750-50p+150p-10p^2\mathrm{....................}[simplify]\\ R=-10p^2+100p+750\end{array}$

So, the function in standard form is :

  $R=-10p^2+100p+750$

(b)

To determine

To Find:

The maximum value of the function.

Answer to Problem 64PPS

The maximum value of the function is 1000.

Explanation of Solution

Given Information:

From part (a) , the standard form of function is :

  $R=-10p^2+100p+750$

Concept Used :

• Quadratic Functions:
• Quadratic functions are non-linear and are of the form:

  $f(x)=a{x}^2+bx+c;a\ne 0$

This is the standard form of Quadratic function.

• Axis of Symmetry: Passes through the vertex and divides the parabola into two congruent halves.

  $x=-\frac{b}{2a}$

• y-intercept = The y coordinate of the point at which the graph intersects y-axis = c
• If a > 0 ,
• The graph of $a{x}^2+bx+c$ opens upward.
• The lowest point on the graph is the minimum.
• If a < 0 ,
• The graph of $a{x}^2+bx+c$ opens downward.
• The highest point on the graph is the maximum.
• The maximum or the minimum is the vertex.

Calculation:

• In the equation $R=-10p^2+100p+750$ ; comparing it with standard form of quadratic equation , we have a = -10 , b = 100 , c = 750 .
• Since, a = -10 < 0 .So, the function opens downward and hence the maximum value of the function occurs at the vertex.
• Find vertex:

The p-coordinate of vertex is

  $\begin{array}{l}p=-\frac{b}{2a}\\ p=-\frac{100}{2(-10)}=\mathrm{5..........................}[a=-10,b=100]\end{array}$

The p-coordinate of the vertex is $p=5$

Substitute the value of p = 5 in the equation $R=-10p^2+100p+750$

• to find the value of R:

  $\begin{array}{l}R=-10p^2+100p+750\\ R=-10{(}^5+100(5)+\mathrm{750.................}[p=5]\\ R=-250+500+750=1000\end{array}$

The vertex is at $(5,1000)$ .

So, the maximum of the function is 1000.

(c)

To determine

To Find:

The price at which the poinsettias should be sold to generate the most sales revenue.

Answer to Problem 64PPS

$10

Explanation of Solution

Given Information:

The marching band is selling poinsettias to buy new uniforms. Last year the band charged $5 each, and they sold 150. They want to increase the price this year, and they expect to lose 10 sales for each $1 increase. The sales revenue R, in dollars, generated by selling the poinsettias is predicted by the function $R=-10p^2+100p+750$ , where p is the number of $1 price increases.

The vertex is (5,1000) and is the maximum of the function.

Concept Used :

• Quadratic Functions:
• Quadratic functions are non-linear and are of the form:

  $f(x)=a{x}^2+bx+c;a\ne 0$

This is the standard form of Quadratic function.

Calculation:

• The vertex is (5,1000) and is the maximum of the function.
• The maximum revenue is $1000 , when p = 5.
• Last year , cost of each poinsettias was $5.
• Now , there are five $1 increases.
• So, total of $5 increase since last year.
• The new price = 5+5 =$10.