Select the correct answer to the following. Manufactured fibers dominate the textile industry because they are versatile and can be tailored to the needs and desires of the consumer.

Select the correct answer to the following. Offshore production is the […].

Fill in the blanks with the correct answers to the following. Check your spelling, grammar, sentence case, capitalization, etc. Only input one term/word per blank. was the first commercially successful synthetic fiber, and was the first manufactured fiber.

The ODE \(y’+4y=t\) has integrating factor \(\rho=e^{4t}\). Multiply both sides by \(\rho\) and write the left side as a single derivative. What do you get?

A population is modeled by \(P(t)=\frac{{12}}{{2+2e^{{-12t}}}}\). Compute \(\lim_{{t\to\infty}}P(t)\). What happens to the population in the long term?

Consider \(y’+y=2e^t\). We’ve already done the integrating factor step for you and rewritten the ODE as $$(e^{t}y)’=2e^{2t}$$ Finish solving for y.

Consider \(y’+3y=6\). We’ve already done the integrating factor step for you and rewritten the ODE as $$(e^{3t}y)’=6e^{3t}$$ Finish solving for y.

The ODE \(y’+3y=e^x\) has integrating factor \(\rho=e^{3x}\). Multiply both sides by \(\rho\) and write the left side as a single derivative. What do you get?

A tank starts with 40 gallons of water. Water flows in at 5 gal/min and out at 2 gal/min. What is V(t)?Here V(t) is the volume of water in the tank (in gallons) at time t (in minutes).

Suppose for a population P(t): the birth rate per person is proportional to P with proportionality constant \(k_1\); the death rate per person is constant and equal to 50. Write the ODE for \(dP/dt\).