Tutor profile: Maia C.
Does the production function Q = 3K^.5 * 2L^2 have constant, decreasing, or increasing returns to scale?
One way to determine if there are increasing, decreasing, or constant returns to scale in a production function is to substitute in a constant value for all inputs, then double that input, and compare the difference in output to the difference in input. For simplicity, let's start with inserting 1 into our equation as both our units of capital and units of labor, This yields: Q = 3(1)^.5 * 2(1)^2, which simplifies to 3(1) * 2(1), or 6, which is our units of output. Now, I'm going to double the units of both labor and capital and use 2 instead of 1, creating the equation Q = 3(2)^.5 * 2(2)^2, which simplifies to 3(1.4142136) * 2(4), which equates to 4.24264 * 8, which comes out to approximately 33.9411, which is Q, or units of output for this equation. Now, when we doubled our input from 1 to 2, we more than doubled our output from 6 to a bit over 33. In fact, the output increased by just over a factor of 5.6 (We can determine this by dividing our second value for Q (33.9411) by our first value of Q (6). Because doubling our units of input more than doubled our units of output, this production function exhibits increasing returns to scale.
What is the participle in the following sentence: 'Looking along the shoreline, Allie noticed several small crabs and seashells.'?
The participle here is "looking", and it is serving as an adjective modifying the noun of this sentence, "Allie". Because our participle ends in '-ing', it is a present participle, as opposed to a past participle, such as "looked".
In the system of equations 3(x) + 5(y) = 38 6(x) + 2(y) = 44 What are the values of x and y?
I'm going to solve this equation using the substitution method. I'll start with 6x + 2y = 44 because I notice that the coefficient of y (2) is a common factor in all of the terms in this equation. So I'll subtract 6x from both sides, leaving the equation: 2y = 44 - 6x. Next, I'll divide both sides of the equation by 2 (2y/2 = (44 - 6x)/2), leaving y = 22 - 3x. Now that I have a single y term isolated, I'll substitute 22 - 3x into our other equation (3x + 5y = 38) in place of y. This will take 3x + 5y = 38, and leave 3x + 5(22 - 3x) = 38. I'll distribute the 5 into the parenthetical term, making the equation read 3x + 110 - 15x = 38, which simplifies to -12x + 110 = 38. Next, I'll subtract 110 from both sides of the equation, leaving -12x = -72. Finally, I'll divide both sides of the equation by -12, leaving x = 6. Now that we know x = 6, that value can be substituted into either of the original equations to find the value of y. I'm choosing to use 6x + 2y = 44. Because I know x = 6, I now have 6(6) +2y = 44, which becomes 36 + 2y = 44. Subtracting 36 from both sides leaves 2y = 8, and dividing both sides by 2 leaves y = 4. Just to double-check I have correct values, I'm going to substitute x= 6 and y = 4 into the other equation given at the beginning of this problem, to make sure the values still work. So 3x + 5y = 38 becomes 3(6) + 5(4) = 38. This simplifies to 18 + 20 = 38, which is correct.
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