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.

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).

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

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

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.

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?

Refer to the International Journal for Traffic and Transport Engineering (Vol. 3, 2013) study of aircraft bird strikes at a Nigerian airport. Recall that an air traffic controller wants to estimate the true percentage of aircraft bird strikes that occur above 100 feet. Determine bow many aircraft bird strikes need to be analyzed in order to estimate the true percentage to within 5% if you use a 95% confidence interval.

Compute $\varphi(n).$ $n=117612=2^{2} \cdot 3^{5} \cdot 11^{2}$

Find y. $y=\left(3+2 x^{3}\right)^{5}$

Verify that the given function is a homomorphism and find its kernel.

$$f: \mathbb{Q}^{*} \rightarrow \mathbb{Q}^{* *}, \text { where } f(x)=|x|$$