3. the school council sells sweatshirts to raise funds. the students sell 500 sweatshirts a year of $45…

3. the school council sells sweatshirts to raise funds. the students sell 500 sweatshirts a year of $45 each. they are planning to decrease the price to generate more sales. an informal survey was taken showing that for every $1 decrease in price they can expect to sell an additional 20 sweatshirts. if the survey results are correct, what price would maximize revenue from sweatshirt sales and what will be the maximum revenue? (hint: revenue = price * quantity) communication 1. give four examples of real life applications of quadratics: 2. compare the factored form and the standard form for a quadratic relation. when would each form be more useful? give a real life example for each.

3. the school council sells sweatshirts to raise funds. the students sell 500 sweatshirts a year of $45 each. they are planning to decrease the price to generate more sales. an informal survey was taken showing that for every $1 decrease in price they can expect to sell an additional 20 sweatshirts. if the survey results are correct, what price would maximize revenue from sweatshirt sales and what will be the maximum revenue? (hint: revenue = price * quantity) communication 1. give four examples of real life applications of quadratics: 2. compare the factored form and the standard form for a quadratic relation. when would each form be more useful? give a real life example for each.

Answer

Explanation:

Step1: Define variables

Let (x) be the number of ($1) decreases in price. The price per sweatshirt is (p = 45 - x). The number of sweatshirts sold is (n=500 + 20x). Revenue (R=(45 - x)(500 + 20x)).

Step2: Expand the revenue function

[ \begin{align*} R&=(45 - x)(500 + 20x)\ &=45\times500+45\times20x-500x - 20x^{2}\ &=22500 + 900x-500x-20x^{2}\ &=- 20x^{2}+400x + 22500 \end{align*} ]

Step3: Find the vertex of the quadratic function

For a quadratic function (y = ax^{2}+bx + c) ((a=-20), (b = 400), (c = 22500)), the (x)-coordinate of the vertex is (x=-\frac{b}{2a}). [x=-\frac{400}{2\times(-20)}=10]

Step4: Find the price and revenue

The price per sweatshirt (p = 45-10=$35). Substitute (x = 10) into the revenue function: [ \begin{align*} R&=-20\times(10)^{2}+400\times10 + 22500\ &=-2000+4000+22500\ &=24500 \end{align*} ]

Answer:

The price that maximizes revenue is ($35) and the maximum revenue is ($24500)