Question

hw 11 - chain rule section 2.6: problem 13 (1 point)
a total - cost function is given by
c(x)=1900(x^{2}+5)^{1.3}+900
where (c(x)) is the total cost, in thousands of dollars, for the production of (x) airplanes. find the rate at which the total cost is changing when 23 airplanes have been sold.
dollars per airplane
round to two decimal places.

Explanation:

Answer:

1. Discipline: Mathematics; Sub - field: Calculus
2. First, find the derivative of \(C(x)\) using the chain rule. If \(y = 1900(u)^{1.3}+900\) and \(u=x^{2}+5\), then \(\frac{dy}{du}=1900\times1.3u^{0.3}\) and \(\frac{du}{dx} = 2x\). By the chain rule \(\frac{dC}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=1900\times1.3\times(x^{2}+5)^{0.3}\times2x\).
3. Evaluate the derivative at \(x = 23\).

• Substitute \(x = 23\) into \(\frac{dC}{dx}\):
• First, calculate \(x^{2}+5=23^{2}+5=529 + 5=534\).
• Then \(\frac{dC}{dx}=1900\times1.3\times534^{0.3}\times2\times23\).
• \(1900\times1.3 = 2470\), \(534^{0.3}\approx6.97\), \(2\times23 = 46\).
• \(\frac{dC}{dx}=2470\times6.97\times46\).
• \(2470\times6.97 = 2470\times(7 - 0.03)=17290-74.1 = 17215.9\).
• \(17215.9\times46=791931.4\).
• Since \(C(x)\) is in thousands of dollars, the rate is \(791931.4\) dollars per airplane. Rounding to two decimal places, the answer is \(791931.40\) dollars per airplane.