Q64. - Algebra 1

Question

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

 

a. Write the function in standard form

b. Find the maximum value of the function.

c. At what price should the poinsettias be sold to generate the most sales revenue? Explain your reasoning.

Step-by-Step Solution

Answer

a. The standard form of the given function is $R=-10p^2+100p+750$.

b. The maximum value of the function is 1000.

c. The poinsettias should be sold at the price of $10 to generate the most sales revenue.

Part a. Step 1. Define function.

A function, in mathematics, is an expression, rule, or law that defines a relationship between one variable (independent variable) and another variable (the dependent variable).

Part a. Step 2. Define the standard form of the quadratic function.

A quadratic function, which is written in the form, $y=a{x}^2+bx+c$, where, $a\ne 0$ is called the standard form of the quadratic function.

Part a. Step 3. Write the function R = 5 + p 150 − 10 p in standard form.

Simplify the function $R=\left(5+p\right)\left(150-10p\right)$ to find the standard form.

$$\begin{gathered} R=\left(5+p\right)\left(150-10p\right) \\ R=5\left(150-10p\right)+p\left(150-10p\right) \\ R=5\left(150\right)-5\left(10p\right)+p\left(150\right)-p\left(10p\right) \\ R=750-50p+150p-10p^2 \\ R=-10p^2+100p+750 \end{gathered}$$

Hence, the standard form of the given function is $R=-10p^2+100p+750$.

Part b. Step 1. Define the standard form of the quadratic function.

A quadratic function, which is written in the form, $y=a{x}^2+bx+c$, where, $a\ne 0$ is called the standard form of the quadratic function.

Part b. Step 2. Define the maximum or minimum point of the function y = a x 2 + b x + c .

The graph of the function $y=a{x}^2+bx+c$

Opens upward and has a minimum value at $x=-\frac{b}{2a}$, when $a>0$.

Opens downward and has a maximum value at $x=-\frac{b}{2a}$,  when $a<0$.

Part b. Step 3. Determine the maximum value of the function of the given function.

From part (a), 

The standard form of the function is $R=-10p^2+100p+750$.

Compare the quadratic function $R=-10p^2+100p+750$ with the standard quadratic function $R=a{p}^2+bp+c$.

$a=-10,b=100,c=750$

Substitute $a=-10$ and $b=100$ in $p=-\frac{b}{2a}$.

$$\begin{gathered} p=-\left(\frac{100}{2\left(-10\right)}\right) \\ p=-\left(\frac{100}{-20}\right) \\ p=-\left(-5\right) \\ p=5 \end{gathered}$$

Since, $a<0$.

So, the graph opens downward and has a maximum value at $p=5$.

Substitute $p=5$ in $R=-10p^2+100p+750$.

$$\begin{gathered} R=-10{\left(5\right)}^2+100\left(5\right)+750 \\ R=-10\left(25\right)+500+750 \\ R=-250+1250 \\ R=1000 \end{gathered}$$

 

Hence, the maximum value of the function is 1000.

Part c. Step 1. Define the standard form of the quadratic function.

A quadratic function, which is written in the form, $y=a{x}^2+bx+c$, where, $a\ne 0$ is called the standard form of the quadratic function.

Part c. Step 2. Define the maximum or minimum point of the function y = a x 2 + b x + c .

The graph of the function $y=a{x}^2+bx+c$

Opens upward and has a minimum value at $x=-\frac{b}{2a}$, when $a>0$.

Opens downward and has a maximum value at $x=-\frac{b}{2a}$,  when $a<0$.       

Part c. Step 3. Determine the price at which the poinsettias be sold to generate the most sales revenue.

The standard form of the function is $R=-10p^2+100p+750$.

Compare the quadratic function $R=-10p^2+100p+750$ with the standard quadratic function $R=a{p}^2+bp+c$.

$a=-10,b=100,c=750$

Substitute $a=-10$ and $b=100$ in $p=-\frac{b}{2a}$.

$$\begin{gathered} p=-\left(\frac{100}{2\left(-10\right)}\right) \\ p=-\left(\frac{100}{-20}\right) \\ p=-\left(-5\right) \\ p=5 \end{gathered}$$

Since, $a<0$.

So, the graph opens downward and has a maximum value at $p=5$.

Last year, the cost of each poinsettias was $5.

Now, there are five $1 increases.

So, the total of $5 increase since last year.

So, the new price $=5+5=\mathrm{\$}10$

 

Therefore, the poinsettias should be sold at the price of $10 to generate the most sales revenue.