Suppliers of a certain brand of digital voice recorders will make 10,000 available in the market if the unit price is $45. At a unit price of $50, 20,000 units will be made available. Assuming that the relationship between the unit price and the quantity supplied is linear, derive the supply equation. Then determine the quantity suppliers will make available when the unit price is $70. (For this problem, I set the equation with the two points (1000,45) & (20000,50)?
And then 50-45/20,000-10,000 which equals to 1/2000 (simplified) as the slope? Or is that only for demand functions?
you are correct with your two points, ignoring the typo.
The slope of the line is (20000-10000)/(50-45) = 2000
So the equation is
y-10000 = 200(x-45)
Now just plug in x=70 to find the desired quantity.
To derive the supply equation, we can use the two points given: (10,000, $45) and (20,000, $50).
The supply equation has the form: Quantity Supplied = m * Unit Price + b
We can use the formula for the slope (m) of a line between two points:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (10,000, $45) and (x2, y2) = (20,000, $50)
m = ($50 - $45) / (20,000 - 10,000)
m = $5,000 / 10,000
m = 0.5
Now we need to find the y-intercept (b) of the line. We can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
Using the point (10,000, $45):
y - $45 = 0.5(x - 10,000)
y = 0.5x - 5,000 + $45
y = 0.5x - 4,955
So the supply equation is: Quantity Supplied = 0.5 * Unit Price - 4,955
Now, we can determine the quantity suppliers will make available when the unit price is $70 by plugging in the value of $70 into the supply equation:
Quantity Supplied = 0.5 * $70 - 4,955
Quantity Supplied = $35 - 4,955
Quantity Supplied = -4,920
Since the quantity supplied cannot be negative, the suppliers will make zero units available when the unit price is $70.
To derive the supply equation, we can use the two points provided: (10000, 45) and (20000, 50). The slope-intercept form of a linear equation is y = mx + b, where m represents the slope and b represents the y-intercept.
First, we can calculate the slope using the formula:
m = (y2 - y1) / (x2 - x1)
m = (50 - 45) / (20000 - 10000)
m = 5 / 10000
m = 0.0005
Now that we have the slope, we can substitute one of the points into the equation and solve for b.
45 = 0.0005 * 10000 + b
45 = 5 + b
b = 45 - 5
b = 40
Therefore, the supply equation is y = 0.0005x + 40.
To determine the quantity suppliers will make available when the unit price is $70, we can substitute the unit price (x) into the supply equation and solve for the quantity (y).
y = 0.0005 * 70 + 40
y = 0.035 + 40
y ≈ 40.035
Rounding up, the quantity suppliers will make available when the unit price is $70 would be approximately 40.