The impulse J due to a force F is the product of the force times the amount of time t for which the force acts. When the force varies over time, $J=\int_{t_{1}}^{t_{2}} F(t) d t$. We can model the force acting on a rocket during launch by an exponential function $F(t)=A e^{b t}$, where A and b are constants that depend on the characteristics of the engine. At the instant lift-oft occurs (t=0), the force must equal the weight of the rocket. (a) Suppose the rocket weighs 25,000 N (a mass of about 2500 kg or a weight of 5500 lb), and 30 seconds after lift-off the force acting on the rocket equals twice the weight of the rocket. Find A and b. (b) Find the impulse delivered to the rocket during the first 30 seconds after the launch.

Construct a (7, 3) code in which every nonzero codeword has Hamming weight at least 4.

Find an equation of the least squares linear regression line of y and x for the given data, and sketch both the line and the data. Predict the value of y corresponding to x=3.5.

$$\begin{array}{ccccccc} x & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline y & 1 & 1.8 & 2 & 4 & 4.5 & 7 & 9 \end{array}$$

Humans and warm-blooded animals maintain a constant body temperature, which is typically higher than their environment. (a) Estimate the maximum efficiency of a thermodynamic engine that is powered by this temperature difference in moderate climate conditions (the living organism is the hot reservoir and the environment is the cold reservoir). (b) Determine the output power of such an engine if the rate at which the energy is transferred from the living organisms to the engine is 100 W.

Find the indefinite integrals. $\int\left(\frac{x}{7}-\frac{3}{4} x^{4}\right) d x$

Let S={0,2,4,6} and T={1,3,5,7}. Determine whether each of the following sets of ordered pairs is a function with domain S and codomain T. If so, is it one-to-one? Is it onto? $a. \{(0,2),(2,4),(4,6),(6,0)\} \\ b. \{(6,3),(2,1),(0,3),(4,5)\} \\ c. \{(2,3),(4,7),(0,1),(6,5)\} \\ d. \{(2,1),(4,5),(6,3)\} \\ e. \{(6,1),(0,3),(4,1),(0,7),(2,5)\}$

Find the integrals by any method. $\int(9 x-6) e^{-30 x+20} d x$

On graph paper, draw seven horizontal line segments that are 24 units in length. Label the endpoints 0 and 1. a. Place each number on the seventh line segment $\frac{1}{12}, \frac{2}{12}, \frac{6}{24}, \frac{1}{8}, \frac{4}{8}, \frac{25}{100}, \frac{2}{10}, \frac{5}{10}, \frac{10}{60}, \frac{6}{12}, \frac{5}{8}, \frac{0}{13}, \frac{11}{11}, \frac{1,000,000}{2,000,000}$. b. Explain why some points have more than one label.

Define a linear operator $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ by

$$T\left(\left[\begin{array}{l} x \\ y \\ z \end{array}\right]\right)=\left[\begin{array}{rr} x+ & 2 z \\ 2 x+y+3 z \\ x-y+3 z \end{array}\right]$$

Determine whether the vector v is in R(T).

$$\mathbf{v}=\left[\begin{array}{l} 1 \\ 3 \\ 0 \end{array}\right]$$

Let $f: R \rightarrow R_{1}$ be an isomorphism of integral domains. Let F be the field of quotients of R and $F_{1}$ the field of quotients of $R_{1}$. Prove that the map $f^{*}: F \rightarrow F_{1}$ given by $f^{*}(a / b)=f(a) / f(b)$ is an isomorphism.

Differentiate the functions. $y=3 x-\frac{\frac{2}{x}-\frac{3}{x-1}}{x-2}$

Our packed bed reactor runs the gas phase reaction $A \rightarrow \mathrm{R}$ at 10 atm and $336^{\circ} \mathrm{C},$ and gives 90% conversion of a pure A feed. However, the catalyst salesman guarantees that in the absence of any pore diffusion resistance our reaction will proceed on his new improved porous catalyst $\left(\mathscr{D}_{e}=2 \times 10^{-6} \mathrm{m}^{3} / \mathrm{m} \text { cat } \cdot \mathrm{s}\right)$ with a rate given by $-r_{\mathrm{A}}^{\prime \prime \prime}=0.88 C_{\mathrm{A}} \frac{\mathrm{mol}}{\mathrm{m}^{3} \mathrm{cat} \cdot \mathrm{s}}$ which is much better than what we now can do. The catalyst is rather expensive, since it is formulated of compressed kookaburra droppings and it is sold by weight. Still, we'll try it when we next replace our catalyst. What diameter of catalyst balls should we order?

Differentiate the functions. $y=e^{2 x^{2}+5}$

The United States Golf Association (USGA) tests all new brands of golf balls to ensure that they meet USGA specifications. One test conducted is intended to measure the average distance traveled when the ball is hit by a machine called "lron Byron." Suppose the USGA wishes to estimate the mean distance for a new brand to within 1 yard with 90% confidence. Assume that past tests have indicated that the standard deviation of the distances Iron Byron hits golf balls is approximately 10 yards. How man y golf balls should be hit by Iron Byron to achieve the desired accuracy in estimating the mean?