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 sh&ws that 3 thousand shirts can be sold at a price of $78 each, and that 7 thousand shirts can be sold at a price of $62 each.

Answer 1

Answer:Based on the historical data, we can determine the linear relationship between the number of shirts sold (n) and the price per shirt (P). Let's use the two data points provided:

Point 1: (3,000 shirts, $78 each)

Point 2: (7,000 shirts, $62 each)

To find the equation of the line, we can use the slope-intercept form: y = mx + b, where y represents the price per shirt (P) and x represents the number of shirts sold (n).

First, let's calculate the slope (m):

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

= ($62 - $78) / (7,000 - 3,000)

= -$16 / 4,000

= -$0.004 per shirt

Now, let's find the y-intercept (b) using one of the data points:

$78 = (-$0.004)(3,000) + b

$78 = -$12 + b

b = $78 + $12

b = $90

Therefore, the equation for the linear relationship between the number of shirts sold (n) and the price per shirt (P) is:

P = -$0.004n + $90

This equation represents the relationship between the number of shirts sold and the price per shirt for the clothing business.

Explanation:

Answer 2

Linear relationship is: P = -0.004n +90

Explanation:

We can use the following linear equation to model the relationship between the number of shirts sold and the price charged per shirt:

P = mn + b

where:

• P is the price charged per shirt
• n is the number of shirts sold
• m is the slope of the line
• b is the y-intercept

We can use the two data points given in the problem to solve for the slope and y-intercept of the line.

(3000, 78)

(7000, 62)

Substituting the first data point into the equation gives us:

78 = m × 3000 + b

78 = 3000m + b

Substituting the second data point into the equation gives us:

62 = m × 7000 + b

62 = 7000m + b

Subtracting the second equation from the first equation gives us:

62 - (78) = 7000m + b - ( 3000m +b)

Open the bracket:

62 - 78 = 7000m + b - 3000m - b

Simplify like terms:

- 16 = 4000m

Divide both sides by 4000.

we get

`\sf (-16)/(4000)=(4000m)/(4000)`

m = -0.004

Substituting this value of m into either of the original equations and solving for b gives us:

78 = - 0.004 × 3000 + b

78 = - 12 + b

isolate b, we get

b = 78 + 12

b = 90

Therefore, the linear equation that models the relationship between the number of shirts sold and the price charged per shirt is:

P = -0.004n +90

We can use this equation to predict the price that the clothing business can charge for any number of shirts.

For example, if the clothing business wants to sell 5000 shirts, we can use the equation to predict the price as follows:

P = -0.004 × 5000 + 90 = 70

Therefore, the clothing business can charge $70 per shirt if it wants to sell 5000 shirts.