On one production line in a small custom chip factory, marginal costs decrease rapidly as more chips are made. The marginal cost of making x thousand chips in a day is thought to be C'(x)=40 / the square root of x in hundreds of dollars. Given that it costs 40,000 dollars to make one thousand chips, find the cost of making 16 thousand chips in a day. Total cost in hundreds would be?
Well, I must say, this sounds like a bit of a math puzzle. Let me put on my thinking cap - or should I say clown hat - and give it a shot!
To find the total cost of making 16 thousand chips in a day, we need to integrate the marginal cost function C'(x)=40/(√x) with respect to x.
Now, let's break it down step by step. Integrating C'(x), we get:
C(x) = ∫(40/(√x)) dx.
To solve this integral, let's use a little trick - the power rule in reverse! By rewriting the square root as a fractional exponent, we get:
C(x) = ∫(40x^(-1/2)) dx.
Now, applying the power rule in reverse, we increase the exponent by one and divide by the new exponent:
C(x) = 40 * (x^(1/2)) / (1/2) + C.
Simplifying further:
C(x) = 80√x + C.
Now, we need to find the value of C when x = 1 (1 thousand chips) given that it costs 40,000 dollars:
40,000 = 80√1 + C,
40,000 = 80 + C,
C = 40,000 - 80,
C = 39,920.
So, our total cost function for making x thousand chips is:
C(x) = 80√x + 39,920.
To find the cost of making 16 thousand chips, we plug x = 16 into the total cost function:
C(16) = 80√16 + 39,920,
C(16) = 80 * 4 + 39,920,
C(16) = 320 + 39,920,
C(16) = 40,240.
So, the cost of making 16 thousand chips in a day would be 40,240 dollars.
But remember, this is in hundreds, so the total cost in hundreds would be 402.4.
Phew! That was quite a clown-tastic math workout! I hope my circus act helped you crack this problem.
To find the cost of making 16 thousand chips in a day, we need to calculate the total cost. The marginal cost is given by the function C'(x) = 40 / √x, where x represents the number of thousand chips made.
To find the total cost, we need to integrate the marginal cost function. The integral of C'(x) with respect to x gives us the total cost function C(x).
Let's integrate C'(x) = 40 / √x:
∫ (40 / √x) dx
Using the power rule of integration, we can rewrite the integral as:
40 ∫ x^(-1/2) dx
Now we integrate x^(-1/2):
40 * (2 * √x) + C
Simplifying, we have:
80 * √x + C
Since the initial cost is $40,000 dollars for 1,000 chips, we can use this information to find the constant C.
80 * √1 + C = 40,000
Simplifying further:
80 + C = 40,000
C = 39,920
So, the total cost function is given by:
C(x) = 80 * √x + 39,920
To find the cost of making 16 thousand chips in a day, we can plug in x = 16 into the total cost function:
C(16) = 80 * √16 + 39,920
C(16) = 80 * 4 + 39,920
C(16) = 320 + 39,920
C(16) = 40,240
Therefore, the cost of making 16 thousand chips in a day is $40,240 dollars.
To find the total cost of making 16 thousand chips in a day, we need to integrate the marginal cost function over the given range.
First, let's determine the indefinite integral of the marginal cost function C'(x) with respect to x:
∫ (40 / √x) dx
To integrate this, we can use the power rule for integration, which states that ∫ x^n dx = (x^(n+1))/(n+1), where n is any real number except -1.
Applying the power rule, we can rewrite the integral as:
40 ∫ x^(-1/2) dx
Now, applying the power rule, we get:
40 * [ (x^(1/2))/(1/2) ] + C
Simplifying further:
80 * √x + C
Given that the marginal cost of making 1,000 chips is $40,000, we can use this information to find the constant C.
When x = 1, the total cost is $40,000:
80 * √1 + C = 40,000
Simplifying this equation, we get:
80 + C = 40,000
C = 39,920
Now, we can find the total cost of making 16 thousand chips.
Plugging x = 16 into our integrated function:
80 * √16 + C
= 80 * 4 + 39,920
= 320 + 39,920
= 40,240
Therefore, the cost of making 16 thousand chips in a day is $40,240 (in hundreds of dollars).