At a price of $9 per box of oranges, the supply is 320,000 boxes and at a price of $8.50 per box, the supply is 270,000 boxes. Find the supply equation of the form p = mq + c, where p is the price in dollars and q is the corresponding supply in thousands of boxes.
p = m q + c
9.0 = 320 m + c
8.5 = 270 m + c
--------------------- subtract
0.5 = 50 m
m = 5/500 = 0.01
then
9 = 320 * 0.01 + c
9 = 3.2 + c
c = 5.8
so
p = 0.01 q + 5.8
Sure, here's the supply equation in the form p = mq + c:
Since the price is $9 per box at a supply of 320,000 boxes, we can plug in these values to find the equation:
9 = m(320) + c
Similarly, at a price of $8.50 per box and a supply of 270,000 boxes, we have:
8.5 = m(270) + c
Now, we have two equations with two unknowns (m and c), which we can solve simultaneously.
First, we'll multiply both sides of the second equation by 2 to eliminate the decimals:
17 = 2m(270) + 2c
Since 270 is divisible by 270, let's divide both equations by 270 to simplify:
1/30 = m + c/270 (first equation)
1/15 = 2m + 2c/270 (second equation)
Now, let's isolate m in the first equation:
m = 1/30 - c/270
Substituting this value of m into the second equation:
1/15 = 2(1/30 - c/270) + 2c/270
1/15 = 2/30 - 2c/270 + 2c/270
1/15 = 1/30 + c/270
To further simplify, let's make the denominators the same:
1/15 = 9/270 + c/270
Combining the fractions on the right side:
1/15 = (9 + c)/270
Now, let's multiply both sides by 270 to get rid of the fraction:
270/15 = 9 + c
18 = 9 + c
Subtracting 9 from both sides:
c = 18 - 9
c = 9
Now that we have the value of c, we can substitute it back into the first equation to find m:
1/30 = m + 9/270
Multiplying both sides by 270:
9 = 270m + 9
Subtracting 9 from both sides:
0 = 270m
Dividing both sides by 270 to solve for m:
0 = m
Therefore, the supply equation in the form p = mq + c is:
p = 0q + 9
In other words, the supply equation is simply p = 9
To find the supply equation, we need to determine the values of m and c in the equation p = mq + c.
Given:
Price at $9 per box (p1) = $9
Supply at $9 per box (q1) = 320,000 boxes
Price at $8.50 per box (p2) = $8.50
Supply at $8.50 per box (q2) = 270,000 boxes
Step 1: Calculate the change in supply (Δq) and change in price (Δp):
Δq = q2 - q1
= 270,000 - 320,000
= -50,000
Δp = p2 - p1
= $8.50 - $9
= -$0.50
Step 2: Calculate the value of m:
m = Δp / Δq
= (-$0.50) / (-50,000)
= 0.01
Step 3: Substitute the values of m, p1, and q1 into the equation to solve for c:
p1 = mq1 + c
$9 = (0.01 * 320,000) + c
$9 = 3,200 + c
Subtracting 3,200 from both sides:
c = $9 - $3,200
c = $5,800
Thus, the supply equation of the form p = mq + c is:
p = 0.01q + 5,800
To find the supply equation of the form p = mq + c, we need to determine the values of m and c.
We are given two points on the supply curve:
1. At a price of $9 per box, the supply is 320,000 boxes.
2. At a price of $8.50 per box, the supply is 270,000 boxes.
Let's assign the first point as (p1, q1) and the second point as (p2, q2):
Point 1: (p1, q1) = ($9, 320)
Point 2: (p2, q2) = ($8.50, 270)
We can now use these two points to solve for m and c.
Step 1: Determine the slope (m)
The slope of the supply equation, m, represents the change in price (Δp) per change in supply (Δq). We can calculate the slope using the formula:
m = (p2 - p1) / (q2 - q1)
Using the values from the given points, we have:
m = ($8.50 - $9) / (270 - 320)
Simplifying this equation:
m = -$0.50 / -50
m = 0.01
Therefore, the slope (m) is 0.01.
Step 2: Determine the y-intercept (c)
The y-intercept (c) represents the price at which the supply curve intersects the y-axis (q = 0). To find c, we can use either of the two points from the given data and substitute the values of p, q, and m into the equation:
p = mq + c
Let's use point 1: (p1, q1) = ($9, 320)
Substituting the values into the equation:
$9 = (0.01)(320) + c
Simplifying this equation:
$9 = 3.20 + c
Subtracting 3.20 from both sides:
$9 - 3.20 = c
$5.80 = c
Therefore, the y-intercept (c) is $5.80.
Step 3: Write the supply equation
Now that we have the values of m and c, we can write the supply equation of the form p = mq + c.
Using the values we found:
m = 0.01
c = $5.80
The supply equation is:
p = 0.01q + $5.80