Please check and correct my answers. Thank you.
Jack Benny can get blood from a stone. If he has x stones, the number of pints of blood he can extract from them is f(x) = 2x^(1/3). Stones cost Jack w dollars each. Jack can sell each pint of blood for p dollars.
a) How many stones does Jack need to extract y pints of blood?
b) What is the cost of extracting y pints of blood?
c) What is Jack’s supply function when stones cost w each? When stones cost $8 each?
d) If Jack has 19 relatives who can also get blood from a stone in the same way, what is the aggregate supply function for blood, Y, when stones cost w dollars each?
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my answers:
a)
y^3/8
b)
(w)(y^3/8)
c)
S(p,w) = (8p/3w)^(1/2)
S(p) = (p/3)^(1/2)
d)
Y = 20(8p/3w)^(1/2)
a) To find the number of stones Jack needs to extract y pints of blood, we can set the equation f(x) = y and solve for x.
2x^(1/3) = y
Taking the cube of both sides, we get:
x = (y/2)^3
Therefore, the correct answer is x = (y/2)^3.
b) The cost of extracting y pints of blood can be found by multiplying the number of stones required (x) by the cost of each stone (w).
Therefore, the correct answer is (w)(x) = (w)((y/2)^3) = (wy^3)/(8).
c) Jack's supply function can be determined by solving the equation p = 2x^(1/3) for x.
For stones costing w each, the correct answer is S(p, w) = (p/w)^(3/2).
For stones costing $8 each, the correct answer is S(p, 8) = (p/12)^(3/2).
d) If Jack has 19 relatives who can also extract blood in the same way, the aggregate supply function for blood, Y, when stones cost w dollars each would be:
Y = 20*(2x^(1/3)) where x is the number of stones.
Since Jack and his 19 relatives all extract blood in the same way, the total number of stones needed would be 20 times the number of stones calculated using the individual supply function.
The correct answer is Y = 20*(2x^(1/3)).
a) To determine how many stones Jack needs to extract y pints of blood, we can set up the equation:
y = 2x^(1/3)
Now, solve for x:
y^3 = 2x
Divide both sides by 2:
x = y^3/2
So, the number of stones Jack needs to extract y pints of blood is given by x = y^3/2.
b) The cost of extracting y pints of blood can be calculated by multiplying the cost per stone, w dollars, by the number of stones needed:
Cost = w * (y^3/2)
Therefore, the cost of extracting y pints of blood is w * (y^3/2).
c) To find the supply function when stones cost w dollars each, we can substitute the given function f(x) = 2x^(1/3) into the equation:
Supply function, S(p,w) = (p/2w)^(3/2)
For the specific case when stones cost $8 each, the supply function becomes:
Supply function, S(p) = (p/16)^(3/2)
So, the supply function is S(p,w) = (p/2w)^(3/2) and S(p) = (p/16)^(3/2) when stones cost $8 each.
d) If Jack has 19 relatives who can also extract blood in the same way, the aggregate supply function for blood, Y, when stones cost w dollars each will be:
Y = 20 * (p/2w)^(3/2)
Here, we multiply by 20 because there are now a total of 20 individuals extracting blood.