A manufacturing company finds that they can sell 375 items at $2.50 per item and 225 items at $3.50 per item. If the relationship between the number of items sold x and the price per item p is a linear one:

Find a formula that gives x in terms of p: x=

Now use the formula to find the number of items they will sell if the price per item is $1.50.
They will sell items if the price is $1.50.

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To find the formula that gives x in terms of p, we need to determine the equation of the line that represents the relationship between the number of items sold and the price per item.

Let's use the two given data points: (375, $2.50) and (225, $3.50).

First, let's find the slope (m) of the line:

m = (change in y) / (change in x)
= (p - 2.50) / (x - 375)

Using the second data point, (225, $3.50):

3.50 - 2.50 = (p - 2.50) / (225 - 375)
1 = (p - 2.50) / (-150)

Cross multiplying:
-150 = p - 2.50

Now, let's solve for p:
p = -150 + 2.50
p = -147.50

The formula that gives x in terms of p is:
x = (-150(p - 2.50)) / (-150)

Now, we can use this formula to find the number of items sold if the price per item is $1.50.

Let's substitute p = $1.50 into the formula:
x = (-150(1.50 - 2.50)) / (-150)
x = (-150(-1.00)) / (-150)
x = 1

Therefore, if the price per item is $1.50, the company will sell 1 item.

To find a formula that gives x in terms of p, we can use the concept of linear equations. In this case, we have two data points:

Data point 1: 375 items sold at $2.50 per item
Data point 2: 225 items sold at $3.50 per item

Let's label the number of items sold as x and the price per item as p. We can create two equations using these data points:

Equation 1: x = 375 when p = 2.50
Equation 2: x = 225 when p = 3.50

To find the equation, we need to find the slope (m) and the y-intercept (b) of the line that represents this linear relationship.

First, find the slope:
m = (y2 - y1) / (x2 - x1)
m = (225 - 375) / (3.50 - 2.50)
m = -150 / 1
m = -150

Now, substitute one of the data points and the slope into the slope-intercept form of a linear equation (y = mx + b) to find the y-intercept (b):

375 = -150(2.50) + b
375 = -375 + b
b = 375 + 375
b = 750

Hence, the linear equation that relates the number of items sold (x) to the price per item (p) is:
x = -150p + 750

Now let's use the formula to find the number of items they will sell if the price per item is $1.50. We can substitute p = 1.50 into the equation:

x = -150(1.50) + 750
x = -225 + 750
x = 525

Therefore, if the price per item is $1.50, the manufacturing company will sell 525 items.

for every $1 price increase, sales decrease by 150. So,

Now you can use the point-slope form of the line:

x-375 = -150(p-2.50)
x = 750 - 150p