Abbey Road Motors pays a percent commission to its sales people. They are paid a percent of the profit the dealership makes on a car. If the profit is under $1,000,
the commission rate is 20%. If the profit is at least $1,000 and less than or equal to
$2,000, the commission rate is 20% of the first $1,000 and 24% of the remainder of
the profit. If the profit is above $2,000, the rate is 20% of the first $1,000 of profit,
24% of the next $1,000 of profit, and 29% of the amount of profit over $2,000. If p
represents the profit, express the commission c(p) algebraically as a split function.

c(p) = {0.2 p ⇒ p < 1000

          0.24 p - 40 ⇒ 1000 ≤ p ≤ 2000

          0.29 p - 140 ⇒ p > 2000}

Step-by-step explanation:

* Lets explain how to solve the problem

- The profit is represented by p

1. If the profit is under $1,000,  the commission rate is 20%

∵ The profit is p < 1000

∵ 20% of p =  = 0.2 p

∵ c(p) is the function of the commission

∴ c(p) = 0.2 p when p < 1000

2. If the profit is at least $1,000 and less than or equal to  $2,000, the

 commission rate is 20% of the first $1,000 and 24% of the remainder

 of  the profit

- At least means greater than or equal

∵ The profit 1000 ≤ p ≤ 2000

- The commission is divided into 20% of first $1000 and 24% of

 the reminder

∵ 20% of 1000 =  = 200

∵ The remainder of the profit = p - 1000

∵ 24% of the remainder profit =

 = 0.24(p - 1000) = 0.24 p - 240

∴ The total commission = 200 + 0.24 p - 240

∴ The total commission = 0.24 p - 40

∴ c(p) = 0.24 p - 40 when 1000 ≤ p ≤ 2000

3. If the profit is above $2,000, the rate is 20% of the first $1,000

  of profit,  24% of the next $1,000 of profit, and 29% of the amount

  of profit over $2,000

∵ The profit p > 2000

- The commission is divided into 20% of first $1000 and 24% of the

 next $1,000 of profit, and 29% of the amount of profit over $2,000

∵ 20% of 1000 =  = 200

∵ 24% of 1000 =  = 240

- The amount of profit over $2,000 = p - 2000

∵ 29% of the amount of profit over $2,000 =

   = 0.29(p - 2000)

  = 0.29 p - 580

∴ The total commission = 200 + 240 + 0.29 p - 580

∴ The total commission = 0.29 p - 140

∴ c(p) = 0.29 p - 140 when p > 2000

* The commission function is:

c(p) = {0.2 p ⇒ p < 1000

           0.24 p - 40 ⇒ 1000 ≤ p ≤ 2000

           0.29 p - 140 ⇒ p > 2000}* Lets explain how to solve the problem

- The profit is represented by p

1. If the profit is under $1,000,  the commission rate is 20%

∵ The profit is p < 1000

∵ 20% of p =  = 0.2 p

∵ c(p) is the function of the commission

∴ c(p) = 0.2 p when p < 1000

2. If the profit is at least $1,000 and less than or equal to  $2,000, the

 commission rate is 20% of the first $1,000 and 24% of the remainder

 of  the profit

- At least means greater than or equal

∵ The profit 1000 ≤ p ≤ 2000

- The commission is divided into 20% of first $1000 and 24% of

 the reminder

∵ 20% of 1000 =  = 200

∵ The remainder of the profit = p - 1000

∵ 24% of the remainder profit =

 = 0.24(p - 1000) = 0.24 p - 240

∴ The total commission = 200 + 0.24 p - 240

∴ The total commission = 0.24 p - 40

∴ c(p) = 0.24 p - 40 when 1000 ≤ p ≤ 2000

3. If the profit is above $2,000, the rate is 20% of the first $1,000

  of profit,  24% of the next $1,000 of profit, and 29% of the amount

  of profit over $2,000

∵ The profit p > 2000

- The commission is divided into 20% of first $1000 and 24% of the

 next $1,000 of profit, and 29% of the amount of profit over $2,000

∵ 20% of 1000 =  = 200

∵ 24% of 1000 =  = 240

- The amount of profit over $2,000 = p - 2000

∵ 29% of the amount of profit over $2,000 =

   = 0.29(p - 2000)

  = 0.29 p - 580

∴ The total commission = 200 + 240 + 0.29 p - 580

∴ The total commission = 0.29 p - 140

∴ c(p) = 0.29 p - 140 when p > 2000

* The commission function is:

c(p) = {0.2 p ⇒ p < 1000

           0.24 p - 40 ⇒ 1000 ≤ p ≤ 2000

           0.29 p - 140 ⇒ p > 2000}

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.