Gator Tee Shirtz has determined that its productivity is given by the equation $f (x, y) = 36 x^{1 / 4} y^{3 / 4}$ , when $x$ units of labor and $y$ units of capital are used. Compute the marginal productivity of capital when $a^{4}$ units of labor and $b^{4}$ units of capital are used where $a$ and $b$ have the following values:
$a = 3, and b = 6 .$
Round your answer to two decimal places.

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So, the marginal productivity of capital when $a^{4}$ units of labor and $b^{4}$ units of capital are used is 27.

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Step 1 ：The productivity function is given by $f (x, y) = 36 x^{1 / 4} y^{3 / 4}$ .

Step 2 ：The marginal productivity of capital is given by the partial derivative of the productivity function with respect to $y$ , denoted as $f_{y}$ .

Step 3 ：We have $f_{y} = 36 x^{1 / 4} * \frac{3}{4} y^{- 1 / 4}$ .

Step 4 ：Substitute $x = a^{4}$ and $y = b^{4}$ into the equation, we get $f_{y} = 36 a * \frac{3}{4} b^{- 1}$ .

Step 5 ：Given $a = 3$ and $b = 6$ , substitute these values into the equation, we get $f_{y} = 36 * 3 * \frac{3}{4} * \frac{1}{6}$ .

Step 6 ：Calculate the above expression, we get $f_{y} = 27$ .

Step 7 ：So, the marginal productivity of capital when $a^{4}$ units of labor and $b^{4}$ units of capital are used is 27.

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