A)For a certain commodity the demand equation is given by D=-3p+20. At the price of $1, four units of the commodity are supplied.If the supply equation is linear and the market price is $4, find the supply equation.
B) how many litres of a 70% alcohol solution must be added to 50 litres of a 40 % alcohol solution to produce a 50% alcohol solution??
3. For a certain commodity the demand equation is given by D=-3p+20.At a price of $1, four units of the commodity are supplied. If the supply equation is linear and the market price is $6, find the supply equation
A) To find the supply equation, we need to use the given information and apply it to the concept of supply and demand equilibrium.
First, let's find the quantity demanded (D) at a price of $1 using the demand equation:
D = -3p + 20
D = -3(1) + 20
D = -3 + 20
D = 17
Since at a price of $1, the quantity supplied is four units, we have:
Qs = 4
Now let's calculate the slope of the supply equation using the given information when the market price is $4. We can use the formula:
slope = (change in quantity supplied)/(change in price)
We have two points:
Point 1: (Qs1, P1) = (4, 1)
Point 2: (Qs2, P2) = (? , 4)
Let's calculate the slope:
slope = (Qs2 - Qs1)/(P2 - P1)
slope = (Qs2 - 4)/(4 - 1)
slope = (Qs2 - 4)/3
Now, we can substitute one of the points into the slope equation to solve for Qs2:
(Qs2 - 4)/3 = 4 - 1
(Qs2 - 4)/3 = 3
Qs2 - 4 = 9
Qs2 = 13
Therefore, at a price of $4, the quantity supplied is 13 units.
Now that we have two points on the supply line: (4, 1) and (13, 4), we can use the point-slope form of a linear equation to find the supply equation. The formula is:
y - y1 = m(x - x1)
Let's choose the first point (4, 1):
Qs - 1 = (Qs2 - 4)/3*(P - 4)
Qs - 1 = (13 - 4)/3*(P - 4)
Qs - 1 = 9/3*(P - 4)
Qs - 1 = 3*(P - 4)
Qs - 1 = 3P - 12
Qs = 3P - 11
Therefore, the supply equation for this commodity is Qs = 3P - 11.
B) To find out how many liters of a 70% alcohol solution must be added to 50 liters of a 40% alcohol solution to produce a 50% alcohol solution, we can set up an equation based on the concept of the mixture.
Let x be the amount (in liters) of the 70% alcohol solution that needs to be added.
The total amount of alcohol before adding is:
0.40 * 50 (from the 40% solution)
After adding x liters of the 70% solution, the total amount of alcohol in the mixture becomes:
0.40 * 50 + 0.70 * x
The total volume of the mixture is:
50 + x
To produce a 50% alcohol solution, the ratio of the total amount of alcohol to the total volume of the mixture should be 0.50.
Setting up the equation:
(0.40 * 50 + 0.70 * x) / (50 + x) = 0.50
Now, we can solve this equation to find the value of x.
(0.40 * 50 + 0.70 * x) = 0.50 * (50 + x)
20 + 0.70x = 25 + 0.50x
0.70x - 0.50x = 25 - 20
0.20x = 5
x = 5 / 0.20
x = 25
Therefore, 25 liters of the 70% alcohol solution must be added to 50 liters of the 40% alcohol solution to produce a 50% alcohol solution.