Brief Calculus.

Question

A clothing business finds there is a linear relationship between the number of sweaters that it can sell and the price they charge per sweater. In particular data shows that when the price was set $30 then 4000 sweaters can be sold. When the price was $34 then 3000 sweaters can be sold.

a. use function notation to write a linear function representing the situation.

b. what is the slope? write a sentence interpreting the meaning of the slope in this scenario.

c. use your function to determine how many sweaters will be sold when the price of a sweater is $21. use a complete sentence to clearly state your answers

d. suppose the company has 7200 sweaters to sell. what price should be charged to sell all 7200 sweaters. use a complete sentence to clearly state your answer.

Answer 1

(3, 34) and (4, 30) = (Q, P) where Q is in units of 1,000, P in dollars

are points on a line, with slope (30-34)/(4-3) = -4

P=-4Q + b, solve for b by plugging in either point

30=-4(4) + b

b =30+16 =46

P=-4Q +46 is an equation describing the linear relation between price and quantity demanded

the slope of -4 means that as price rises by $4, quantity purchased declines by 1,000 or if price decreases by $4, quantity purchased increases by 1,000 sweaters

If price were $21 6,250 sweaters will be purchased

21 =-4Q +46

4Q = 25

Q = 25/4 = 6,250 thousand sweaters sold

P=-4(7.2) + 46

P = -28.8 + 46 = $17.20 for the price of a sweater to sell 7,200 sweaters