Question

Dave owns Dave's Donut Shop and must make decisions about how much to charge for donuts. As a businessowner he wants to attract a lot of customers as well as make a lot of money (revenue). The amount of revenue Dave makes can be modeled by the equation R = (p-1)(200 - 40p), where p represents the price of one donut and R represents the total revenue. Use as many strategies as you can to figure out the questions below. 0 At what price will Dave break even (make no revenue)? How much should Dave charge to maximize his revenue? What is the most money Dave can make? Dave made $100 from donut sales. How much must he have charged per donut?​

Answer 1

Answer:

2430

Step-by-step explanation:

Answer 2

The revenue is the amount of money generated by an organization from sales or services made in a specific time. The conclusion from the computation made are:

• The break even price is at $1 and $5

• To maximize his revenue, he as to charge $3

• The most money he can make is $160

• He charged $4.225 or $1.775 to make $100.

Given that:

$R = (p - 1)(200 - 40p)$

To calculate the break even, we set R to 0.

So, we have:

$(p - 1)(200 - 40p) =0$

Split

$p - 1 = 0$ or $200 - 40p = 0$

Solve for p

$p = 1$ or $40p = 200$

Divide by 40 in the second equation

$p=1$ or $p =5$

The break even price is at $1 and $5

To maximize his revenue, we have:

$R = (p - 1)(200 - 40p)$

Open bracket

$R = 200p - 200 - 40p^2 + 40p$

Rewrite as:

$R = - 40p^2+ 200p + 40p- 200$

$R = - 40p^2+ 240p- 200$

Differentiate

$R' = -80p + 240$

Equate to 0 and solve for p

$-80p + 240=0$

$-80p =- 240$

Divide by -80

$p = 3$

To maximize his revenue, he as to charge $3

The most money he can make is calculated as:

$R = (p - 1)(200 - 40p)$

Substitute $p = 3$

$R = (3 - 1)(200 - 40 \times 3)$

$R = (3 - 1)(200 - 120)$

$R = 160$

The most money he can make is $160

The charges he sold when he made $100 is:

$R = - 40p^2+ 240p- 200$

Substitute 100 for R

$- 40p^2+ 240p- 200=100$

Collect like terms

$- 40p^2+ 240p- 200-100=0$

$- 40p^2+ 240p- 300=0$

Divide through by -20

$2p^2- 12p+ 15=0$

Using quadratic formula, the solution of p is:

$p = \frac{-b \± \sqrt{b^2 - 4ac} }{2a}$

So, we have:

$p = \frac{-(-12) \± \sqrt{(-12)^2 - 4\times 2 \times 15} }{2 \times 2}$

$p = \frac{12 \± \sqrt{24} }{4}$

$p = \frac{12 \± 4.90 }{4}$

Split

$p = \frac{12 + 4.90 }{4}\ or p = \frac{12 - 4.90 }{4}$

$p = \frac{16.90 }{4}\ or\ p = \frac{7.1 }{4}$

$p = 4.225\ or\ p = 1.775$

He charged $4.225 or $1.775 to make $100.

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