Question

(15/100) (Supply theory) 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) = 2x¹/³ Stones cost Jack Sw each. Jack can sell each pint of blood for Sp. How many stones does Jack need to extract y pints of blood?

Answer 1

To find the number of stones Jack needs to extract y pints of blood, one must integrate the derivative of the blood extraction function, f'(x) = 2x^(1/3), yielding the function f(x) = 3x^(4/3). Setting f(x) equal to y and solving for x, the number of stones required is (y/3)^(3/4).

Step-by-step explanation:

The question involves finding out how many stones Jack needs to extract y pints of blood, given that the rate of blood extracted from the stones can be described by the function f'(x) = 2x1/3. We are provided with a derivation function, indicating that to find the total amount of blood f(x) extracted from x stones, we'll need to integrate the given function f'(x).

To find f(x), integrate f'(x):

1. ∫ f'(x) dx = ∫ 2x1/3 dx.
2. The antiderivative of 2x1/3 is (3/2)(2)x4/3, which simplifies to 3x4/3.
3. To extract y pints of blood, set the integrated function equal to y: 3x4/3 = y.
4. Solve for x to find the number of stones needed: x4/3 = y/3, and then x = (y/3)3/4.

Therefore, the number of stones Jack needs to extract y pints of blood is given by (y/3)3/4 stones.