A manufacturing company finds that they can sell 300 items at $3.50 per item and 500 items at $ 1.50 $1.50 per item. If the relationship between the number of items sold x x and the price per item p 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 if the price per item is $2.50.
To find a formula that gives the relationship between the number of items sold (x) and the price per item (p), we can use the information provided:
Let's use the point-slope form of a linear equation:
p - p₁ = m(x - x₁),
where p₁ and x₁ are the coordinates of a point on the line, m is the slope of the line, and p and x are the variables.
We are given two points: (300, $3.50) and (500, $1.50). We can use these points to find the slope (m):
m = (p₂ - p₁) / (x₂ - x₁)
= ($1.50 - $3.50) / (500 - 300)
= -$2 / 200
= -0.01
Now that we have the slope, we can choose any point and substitute it into the point-slope equation to find the formula for x in terms of p.
Let's use the point (300, $3.50):
$3.50 - $3.50 = -0.01(x - 300)
0 = -0.01x + 3
Simplifying the equation, we get:
0.01x = 3
x = 300
Therefore, the formula for x in terms of p is:
x = -0.01p + 300.
To find the number of items they would sell if the price per item is $2.50, we substitute p = $2.50 into the formula:
x = -0.01($2.50) + 300
x = -0.025 + 300
x ≈ 299.975.
Rounding to the nearest whole number, the number of items they will sell if the price per item is $2.50 is approximately 300.
To find a formula that gives the number of items sold (x) in terms of the price per item (p), we can use the concept of linear equations.
Given that the relationship between x and p is linear, we can assume it follows the equation of a line: y = mx + b, where y represents the price per item and x represents the number of items sold.
Let's assign the price per item to the y-axis and the number of items sold to the x-axis. We have two points: (300, 3.50) and (500, 1.50), which gives us two equations:
1. 3.50 = m * 300 + b
2. 1.50 = m * 500 + b
We can solve this system of equations to find the values of m and b.
First, let's subtract equation 2 from equation 1:
3.50 - 1.50 = m * 300 + b - m * 500 - b
Simplifying:
2 = m * 300 - m * 500
Factoring out m:
2 = m * (300 - 500)
Simplifying further:
2 = m * (-200)
Dividing both sides by -200:
m = -2/200
m = -0.01
Now, substitute the value of m into equation 1:
3.50 = -0.01 * 300 + b
Simplifying:
3.50 = -3 + b
Adding 3 to both sides:
6.50 = b
So, we have the values m = -0.01 and b = 6.50. Therefore, the formula that gives the number of items sold (x) in terms of the price per item (p) is:
x = -0.01p + 6.50
To find the number of items they will sell if the price per item is $2.50, substitute p = 2.50 into the formula:
x = -0.01 * 2.50 + 6.50
Simplifying:
x = -0.025 + 6.50
x ≈ 6.475
Therefore, if the price per item is $2.50, the company will sell approximately 6.475 items.
You have two ordered pairs, (300 , 3.5) and (500 , 1.5)
find the slope m and use the equation,
p = mx + b