The cost of producing d hundred plastic toy dinosaurs per day in a small company
is C(d) = 120 + 30d + 1
8d4 dollars. This company is currently producing 500 di-
nosaurs each day. Using calculus, estimate how much the company should change the
production to make their daily cost approximately 250 dollars.
do i take the derivative of C(d)? i am so confused
using Calculus ?????
150 = 120 + 30d + 1
since we are estimation
30 = appr 30d
d = 1 ??????
This question makes no sense to me the way you typed it
what is 8d4 dollars ??
C(d) = 120 + 30d + 1/8 d^4 sorry the formating gets off
I'm also confused on where the calculus comes in. Currently,
C(5) = 348.125
We want
120 + 30d + 1/8 d^4 = 250
d=3.62
So, making 362 dinos will cost $250
Solving quartics is not easy, but maybe you're supposed to bring in calculus via the Newton-Raphson method.
Also, this factory is strange, having a production cost that rises as d^4!
Yes, to solve this problem using calculus, you will need to take the derivative of the cost function C(d) with respect to d. Let's go step by step to find the derivative and solve the problem.
Step 1: Start with the given cost function C(d) = 120 + 30d + 1/(8d^4).
Step 2: Take the derivative of C(d) with respect to d. To do this, you will need to use the power rule and the chain rule.
The power rule states that if you have a function f(x) = x^n, the derivative of f(x) with respect to x is given by f'(x) = nx^(n-1).
The chain rule states that if you have a composite function f(g(x)), then the derivative of f(g(x)) with respect to x is given by f'(g(x)) * g'(x).
In our cost function C(d), we have three terms: 120, 30d, and 1/(8d^4). Taking the derivative of each term separately:
- The derivative of 120 is 0 because it is a constant term.
- The derivative of 30d is 30 because we use the power rule (n = 1) and d^(n-1) = d^(1-1) = d^0 = 1.
- The derivative of 1/(8d^4) involves both the power rule and the chain rule. Applying the power rule, we get -4/(8d^(4-1)) = -4/(8d^3) = -1/(2d^3). Then, applying the chain rule, we multiply by the derivative of the exponent, which is -3d^(-3-1) = -3d^(-4) = -3/(d^4).
So, the derivative of C(d) with respect to d is C'(d) = 30 - 1/(2d^3) - 3/(d^4).
Step 3: Plug in the given value of d = 500 into the derivative equation C'(d) = 30 - 1/(2d^3) - 3/(d^4).
C'(500) = 30 - 1/(2 * 500^3) - 3/(500^4).
Simplify the expression to get the estimated change in production required to make the daily cost approximately $250.