Question

A clothing business finds there is a linear relationship between the number of shirts, n it can sell and the price, P, it can charge per shirt. In particular, historical data shows that 3 thousand shirts can be sold at a price of $30 each, and that 6 thousand shirts can be sold at a price of $15 each.

Find the equation of the form P(n)=mn+b that gives the price P they can charge for n thousand shirts.

P(n)=−5n+30

P(n)=−3n+39

P(n)=−9n+57

P(n)=−9n+69

P(n)=−5n+45

P(n)=−3n+33

Answer 1

P(n) = -5n + 45

Explanation:

Use the given information as two points of the linear function.

(3000, 30), (6000, 15)

Now we change it to thousands of shirts

(3, 30), (6, 15)

P(n) = mn + b

m = (15 - 30)/(6 - 3)

m = -15/3

m = -5

P(n) = -5 × n + b

30 = -5 × 3 + b

30 = -15 + b

b = 45

P(n) = -5n + 45

Answer 2

Answer: To find the equation of the form P(n) = mn + b, we need to find the slope (m) and the y-intercept (b) of the line that relates the number of shirts sold (n) to the price charged per shirt (P).

We are given two points on this line: (3, 30) and (6, 15). Using the formula for the slope of a line passing through two points:

m = (P2 - P1) / (n2 - n1)

where (n1, P1) = (3, 30) and (n2, P2) = (6, 15)

m = (15 - 30) / (6 - 3)

m = -5

So the slope of the line is -5.

To find the y-intercept, we can use the point-slope form of a line:

P - P1 = m(n - n1)

where (n1, P1) = (3, 30) and m = -5

P - 30 = -5(n - 3)

P - 30 = -5n + 15

P = -5n + 45

So the equation of the line is P(n) = -5n + 45, which is of the form P(n) = mn + b as required. Therefore, the answer is (E) P(n) = -5n + 45.

Explanation: