A large factory knows that if it sells it's new gadgets for $10 each, it can sell 350 per month and if it sells the same gadgets for $8 it will sell 500 per month assuming the relationship between price and sales is linear predict the monthly sales of gadgets to the nearest whole number if the price is $6.
Im really confused
if the relationship is linear, the line contains the points
(10,350) and (8,500)
so, the equation of the line is
y-350 = -75(x-10)
so, when x=6,
y = -75(6-10)+350 = 650
To predict the monthly sales of gadgets when the price is $6, we need to determine the relationship between price and sales using the given data points. Since the relationship between price and sales is assumed to be linear, we can use the concept of slope and intercept to find the equation of the linear relationship.
Let's consider the two data points - when the price is $10, the factory sells 350 gadgets, and when the price is $8, it sells 500 gadgets.
We can use the slope formula to calculate the slope (m) of the line:
m = (y2 - y1) / (x2 - x1),
where (x1, y1) represents the first data point and (x2, y2) represents the second data point.
x1 = 10 (price in dollars)
y1 = 350 (monthly sales)
x2 = 8 (price in dollars)
y2 = 500 (monthly sales)
Using the formula, we can calculate the slope:
m = (500 - 350) / (8 - 10)
m = 150 / -2
m = -75
Now we have the slope (m) of the linear relationship. Let's find the equation of the line in the form y = mx + b, where b represents the y-intercept.
We can pick one of the data points to substitute into the equation and solve for b. Let's use the point (10, 350):
350 = -75 * 10 + b
350 = -750 + b
b = 350 + 750
b = 1100
Therefore, the equation of the line is y = -75x + 1100.
Now, let's substitute the price $6 (x = 6) into the equation to predict the monthly sales (y):
y = -75 * 6 + 1100
y = -450 + 1100
y = 650
Therefore, when the price is $6, the factory is predicted to sell approximately 650 gadgets per month.
To predict the monthly sales of gadgets when the price is $6, we need to use the information given about the relationship between price and sales being linear.
Let's set up a linear equation using the information we have. We know that at a price of $10, the factory sells 350 gadgets per month. We can represent this as the point (10, 350), where the x-coordinate represents the price and the y-coordinate represents the sales.
Similarly, at a price of $8, the factory sells 500 gadgets per month, which can be represented as the point (8, 500).
We can use these two points to find the equation of the line. The equation of a line in slope-intercept form is y = mx + b, where m represents the slope of the line and b represents the y-intercept. To find the slope (m), we use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line.
Using the points (10, 350) and (8, 500), we can calculate the slope:
m = (500 - 350) / (8 - 10) = 150 / -2 = -75
Now that we have the slope, we can find the y-intercept (b) by substituting one of the known points into the equation y = mx + b. Let's use the point (10, 350):
350 = -75(10) + b
350 = -750 + b
b = 350 + 750
b = 1100
So the equation of the line is y = -75x + 1100.
Now we can determine the sales when the price is $6 by substituting x = 6 into the equation:
y = -75(6) + 1100
y = -450 + 1100
y = 650
Therefore, when the price is $6, the factory can predict monthly sales of 650 gadgets to the nearest whole number.