The manager of Dixie Furniture Company estimates that the daily output of her fa
ID: 2495160 • Letter: T
Question
The manager of Dixie Furniture Company estimates that the daily output of her factory (in thousands of dollars) Q is given by Q = 2x^1/4y^3/4 where x is the amount spent on labor and y is the amount spent on capital (both measured in thousands of dollars).
(a) Find the daily output of the factory if $81,000 is spent on labor and $256,000 is spent on capital each day.
(b) Suppose that the output of the factory is to be maintained at the level found in part (a). By how much should the amount spent on capital be changed if the amount on labor is increased by $1000? (Use the differential to calculate the answer. Round your answer to the nearest cent.)
What is the MRTS? (Round your answer to the nearest cent.) per thousands of dollars
Explanation / Answer
Q = 2x1/4y3/4
(a) x = 81,000 and y = 256,000
Q = 2 x (81,000)1/4 x (256,000)3/4 = 2 x 16.87 x 11,381 = 383,995
(b) x = 81,000 + 1,000 = 82,000
383,995 = 2 x (82,000)1/4 x y3/4
383,995 = 2 x 16.92 x y3/4
y3/4 = 11,347.37
y = 254,992.23
Change in y = 256,000 - 254,992.23 = 1,007.77
(c) MRTS = MPx / MPy
MPx = dQ / dx = 2. (y / x)3/4
MPy = dQ / dy = 2. (x / y)1/4
MRTS = y / x = 256,000 / 81,000 = 3.16
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