Mathematics Questions and Answers

A circular coil 17.0 cm in radius and composed of 145 tightly wound turns carries a current of 2.30 A in the counterclockwise direction, where the plane of the coil makes an angle of 15.0° with the y axis (see figure below). The coil is free to rotate about the z axis and is placed in a region with a uniform magnetic field given by $$ \vec{B} = 1.25 \, \vec{f} \, t $$. (a) What is the magnitude of the magnetic torque on the coil? $$ \mu = 1 \, \text{N} \times \text{m} $$ (b) In what direction will the coil rotate? - clockwise as seen from the +z axis - counterclockwise as seen from the +z axis

# MECE 3338: Funwork 10 – Due Date Monday 11/13/2023 11:59 PM ## Problem 1: Consider the following system that consists of an inverted pendulum OB with mass **m = 6 kg** and length **2α** where **α = 0.5 m**. The pendulum rotates about its bottom end point O with rotational viscous friction coefficient **b_r = 8 N m s/rad** and is supported by two horizontal springs of equal stiffness **k = 100 N/m** attached at its middle point C, as shown. A horizontal spring of same stiffness **k = 100 N/m** is attached at the pendulum's top point **D**. A **displacement input z(t)** at the free end point A of the spring sets the system to motion. Small angular displacements from the vertical equilibrium position of the pendulum are considered. a) Derive the transfer function **G(s)** of the system from the **displacement input z(t)** to the **angular displacement output θ(t)** of the pendulum. Use the values of the system parameters **α, m, k** and **b_r** as stated above in the calculation of **G(s)**. b) Considering a **sinusoid displacement input z(t) = A sin(αt)** find the **worst-case excitation frequency α'** that results in the **maximum steady-state angular displacement amplitude |θ|_max** of the pendulum from its vertical equilibrium position. c) Find this **maximum angular displacement amplitude |θ|_max** if the amplitude of the sinusoid displacement input excitation is **A = 0.1 m**. d) Sketch the **magnitude Body diagram** of the transfer function **G(s)** from the displacement input **z(t)** to the angular displacement output **θ(t)** of the pendulum. e) We want to reduce the steady-state angular displacement amplitude **|θ|_max** to half of the value calculated in c), that is, **|θ'| = |θ|_max / 2**, by changing the rotational viscous friction coefficient to a new value **b_r**. What should be this **new value of the rotational viscous friction coefficient b_r'** such that at the excitation frequency **α'** calculated in b) **the steady-state angular displacement amplitude is reduced by half?** Fig (1): Vertical pendulum with pulling spring on top

) Graph the function f ( x ) = - 2 s i n ( 2 x ) + 3 . a ) Draw a dotted line for the midline on the graph b ) Mark a point for the beginning and end of one period of the function c ) Mark a point for the maximum point d ) Mark a point for the minimam point e ) Draw the graph connecting these points. Double check that this is the graph of the function. Double click the graph below to open and edit in Paint. 2 . ) Using the graphed function above, find the following: \ table [ [ Maximum: , Minimum: ] , [ Equation of the midline:,Amplitude: ] , [ Period: , Frequency: ] , [ Equation of the graphed, ] ] Name: Date: Graded Assignment: Unit 2 Test, Part 2: Graphs of Sinusoidal Functions Answer the questions below. All answers must be handwritten, and all work must be shown to earn credit. No DESMOS graphs will be accepted as answers. 1.) Graph the function $f(x)=-2 \sin (2 x)+3$. (S pts.) You must graph at least one full period. Follow the steps below to graph. a) Draw a dotted line for the midline on the graph b) Mark a point for the beginning and end of one period of the function c) Mark a point for the maximum point d) Mark a point for the minimum point e) Draw the graph connecting these points. Double check that this is the graph of the function. Double click the graph below to open and edit in Paint. 2.) Using the graphed function above, find the following: (S pts.) Double click the star to open and edit with the PAINT program. | Maximum: | Minimum: | | :-- | :-- | | | | | Equation of the midline: | Amplitude: | | | | | Period: | Frequency: | | | | Equation of the graphed function:

What slope is perpendicular to m=5/8?

A corporation must appoint a president, chief executive officer ( CEO ) , chief operating officer ( COO ) , and chief financial officer ( CFO ) . It must also appoint a planning committee with three different members. There are 1 3 qualified candidates, and officers can also serve on the committee. Complete parts ( a ) through ( c ) below. a . How many different ways can the four officers be appointed? There are different ways to appoint the four officers. b . How many different ways can a committee of three be appointed? There are different ways to appoint a committee of three. c . What is the probability of randomly selecting the committee members and getting the three youngest of the qualified candidates? P ( getting the three youngest of the qualified candidates ) = ( Type an integer or a simplified fraction. )

Plot four points on the graph of the function: the *leftmost point*

Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = -3 and g'(5) = 6 Enter your answer in terms of y and x.

write all the possible factors of the following (i) 8x² (ii) 17xy² (iii) 33yz (iv) -6ab class 8 write all the possible factors of the following (i) 8x² (ii) 17xy² (iii) 33yz (iv) -6ab class 8

Solve the following ordinary differential equations. $$ \begin{aligned} & x \frac{d y}{d x}-k y=x^{2} \\ & \frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}-2 y=0 \\ & \frac{d^{2} y}{d x^{2}}+k^{2} y=\sin x \quad\left(k^{2} \neq 0,1\right) \\ & \frac{d^{4} y}{d x^{4}}-k^{4} y=0 \quad \text { (vibration of a beam) } \\ & \text {.. } \\ & \frac{d^{4} y}{d x^{4}}-2 k^{2} \frac{d^{2} y}{d x^{2}}+k^{4} y=0 \quad \text { (bending of an elastic plate) } \\ & \frac{d^{2} y}{d x^{2}}+y=\sec x \end{aligned} $$

# Which nuclear equation represents beta decay? 1. $\quad{ }_{13}^{27} \mathrm{Al}+{ }_{2}^{4} \mathrm{He} \rightarrow{ }_{15}^{30} \mathrm{P}+{ }_{0}^{1} \mathrm{n}$ 2. $\quad{ }_{92}^{238} \mathrm{U} \rightarrow{ }_{90}^{234} \mathrm{Th}+{ }_{2}^{4} \mathrm{He}$ 3. $\quad{ }_{89}^{227} \mathrm{Ac} \rightarrow{ }_{90}^{227} \mathrm{Th}+{ }_{-1}^{0} \mathrm{e}$ 4. $\quad{ }_{18}^{40} \mathrm{Ar} \rightarrow{ }_{19}^{40} \mathrm{~K}+{ }_{-1}^{0} \mathrm{e}$

Which graph matches the equation y+3=2(x+3)?

"Graph of P & S Wave Travel Times, showing Travel time in minutes across distance in kilometers for P and S waves. S wave starts at zero kilometers, zero minutes, and increases to 10,000 kilometers, 25 minutes. P wave starts at zero kilometers, zero minutes and increases to 10,000 kilometers, 13 minutes. About how far does the S wave travel through Earth in 13 minutes? A. 2,000 km B. 4,000 km C. 6,000 km D. 8,000 km"

A ALEKS - ... C (A) A Initial Knowledge Check Add. (-7)/(9y^(2))+(3)/(2y) Simplify your answer as much as possible.

Exercise 4.6.5. [Used in Exercise 7.4.1.] Let $(a, b) \subseteq \mathbb{R}$ be a non-degenerate open bounded interval, and let $f:(a, b) \rightarrow \mathbb{R}$ be a function. Suppose that $f$ is continuous, strictly increasing and bounded. Let $F:[a, b] \rightarrow \mathbb{R}$ be defined by $$ F(x)= \begin{cases}\operatorname{glb} f((a, b)), & \text { if } x=a \\ f(x), & \text { if } a<x<b \\ \operatorname{lub} f((a, b)), & \text { if } x=b\end{cases} $$ (1) Prove that $F$ is continuous.

calculate the break-even point in units use the formula: Break-Even point (units) = Fixed Costs ÷ (Sales price per unit – Variable costs per unit) or in sales dollars using the formula: Break-Even point (sales dollars) = Fixed Costs ÷ Contribution Margin. Suppose there are TWO locations for lunches: One in a higher-income area, and the other in a lower-income area. A: Fixed Expense = \$5,000. Revenue/unit=\$15. Cost/unit=\$5 B: Fixed Expense = \$6,000. Revenue/unit=\$13. Cost/unit=\$5 Compute breakeven quantity separately for A and B. [Post calculations and your answer] Combine $A$ and $B$ to compute combined breakeven. [Post calculations and your answer] Does it look like B is getting a subsidy from A? Is there a different word to use other than 'subsidy'?

Express 105 km/h in miles per hour. Round to the nearest hundredth.

Is 96 divisible by 16 ? - ☐ Yes, because 96 = 16 · 7. - ☐ Yes, because 6 = $$\frac{16}{96}$$. - ☐ Yes, because 96 = 16 · 6. - ☐ No, because 16 is not a divisor of 96. - ☐ No, because 96 is not a multiple of 16.

"What is the mass in kilograms of an individual who weighs 197 lbs.? A. 197 kg B. 8.95 kg C. 89.5 kg D. 90 kg E. 433 kg"

How do you calculate 4x to the second power?

Find the simplified product where $x \geq 0: \sqrt{5 x}\left(\sqrt{8 x^{2}}-2 \sqrt{x}\right)$ - $\sqrt{10 x}$ - $2 x \sqrt{40 x}-2 x$ - $2 x \sqrt{10 x}-2 \sqrt{5 x}$ - $2 x \sqrt{10 x}-2 x \sqrt{5}$

Replace **?** with an expression that will make the equation valid. $$\frac{d}{dx}(9x + 2)^5 = 5(9x + 2)^4 \quad ?$$ The missing expression is ☐. Replace **?** with an expression that will make the equation valid. $$\frac{d}{dx} e^{x^5 + 8} = e^{x^5 + 8} \quad ?$$ The missing expression is ☐. Replace **?** with an expression that will make the equation valid. $$\frac{d}{dx} \ln \left(x^5 + 8\right) = \frac{1}{x^5 + 8} \quad ?$$ The missing expression is ☐.

How many triangles are there? ( $97.67 \%$ get this wrong)

Find the area of the shaded region below.

11. (draw the skeletal structure) You are given an alkene in the assignment (trans-hex-3-ene) Treat this alkene with peroxyacetic acid. Before submitting your structure, make sure you have followed the input format for showing the stereochemistry of rings. You need to make sure the tapered end of the wedge or dashed bond is connected to the ring. You should not be drawing the carboxylic acid by-product.

What are the four steps to solving an equation?

Determine the sum of the following series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{5^{n}} $$

Calculate the total number of gumballs that need to fit in the machine if each gumball has a diameter of 1.25 inches.

Identify the highlighted structure and all of numerous parallel lines around it.

Use the Definition to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit. $$ f(x)=x^{2}+\sqrt{1+2 x}, \quad 6 \leq x \leq 8 $$ $\lim _{x \rightarrow \infty} \sum_{i=1}^{8} \square$ # Question 2 The area $A$ of the region $S$ that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles. $$ A=\lim _{x \rightarrow \infty} R_{n}=\lim _{x \rightarrow \infty}\left[f\left(x_{1}\right) \Delta x+f\left(x_{2}\right) \Delta x+\ldots+f\left(x_{n}\right) \Delta x\right] $$ Use this definition to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit. $$ f(x)=\frac{8 f(x)}{x}, 6 \leq x \leq 11 $$ $A=\lim _{x \rightarrow \infty} \sum_{i=1}^{8} \square$ ## Example with answer The area $A$ of the region $S$ that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles. $$ A=\lim _{x \rightarrow \infty} R_{n}=\lim _{x \rightarrow \infty}\left[f\left(x_{1}\right) \Delta x+f\left(x_{2}\right) \Delta x+\ldots+f\left(x_{n}\right) \Delta x\right] $$ Use this definition to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit. $$ f(x)=\sqrt[3]{x}, 1 \leq x \leq 12 $$ $A=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \quad \sqrt{\sqrt{1+\frac{11 i}{n}} \frac{11}{n}}$ The answer were not.

"What is the distance between the points (−5,−9) and (−5,13)? a. 4 units b. 11 units c. (no value provided) d. 22 units"

Which is the smaller unit in size, a centi- or a milli- metric unit?

"Use function notation to write a recursive formula to represent the sequence: 3, 6, 9, … A. f(n)=f(n−1)+3 B. f(n)=f(n−1)+2 C. f(n)=f(n−1)⋅3 D. f(n)=f(n−1)⋅2"

# EXERCISE 29 PROBLEMS-PART I The following questions are based on the lower right (southeast) corner of the "Greasewood Spring, Arizona," quadrangle (Maps T-18 and T-19 shown in color in the back of the Lab Manual): 1. What is the contour interval of the map? $\qquad$ feet 2. What is the difference in elevation between index contours? $\qquad$ feet 3. What is the elevation of the highest contour line at the top of the eastern Twin Butte? $\qquad$ feet 4. What do the dashed blue lines represent? $\qquad$ 5. What do the gray lines around and between the Twin Buttes represent? $\qquad$ 6. What is the name of the adjoining quadrangle to the south? 7. The latitude of the upper right (northeast) corner of the map is $35^{\circ} 30^{\prime} \mathrm{N}$, while the longitude of the lower left (southwest) corner of the map is $110^{\circ} 00^{\prime} \mathrm{W}$. Why is this called a " 7.5 minute" topographic map? (Hint: What is "minute" referring to?) 8. At the time this quadrangle was printed, what was the difference (in degrees) between true north and magnetic north for this map? $\qquad$ degrees 9. On all three graphic scales for the map, " 0 " is not at the far left. Explain the reason for this. 10. Using the graphic map scales, determine the maximum width of the eastern Twin Butte (use the $6100^{\prime}$ contour to be the outer edge of the butte). $\qquad$ feet kilometers

2. A radio station is giving away tickets to a play. They plan to give away tickets for seats that cost $\$ 10$ and $\$ 20$. They want to give away at least 20 tickets. The total cost of all the tickets they give away can be no more than $\$ 280$. Use a graph to determine possible outcomes.

"What is the domain of the function given the points: (9, -2), (-2, -10), (5, -9), and (4, 5)? "

A speed of 150 km/hr is equivalent to how many miles per hour?

Which form is used to record combinations of security containers?

Let point $C$ be between $V$ and $W$ on $\overline{\text { FW }}$. Given that $V W=61, V C=z+13$, and $C W=z+8$, solve for $z$. A 18 B 19 C 20 D 21 Please select the best answer from the choices provided A B C D

"A speed of 30 miles per hour (mph) is equivalent to 48 kilometers per hour (km/h). Jane passes a sign stating that the speed limit is 50 km/h. What is the speed limit in miles per hour? The speed limit is _____ mph. (Round to the nearest hundredth as needed.)"

"Which description is represented by a discrete graph? A. Kiley bought a platter for $19 and several matching bowls that were $8 each. What is the total cost before tax? B. The temperature at 9 a.m. was 83°F and is heating up at an average rate of 6°F per hour. What is the temperature x hours later? C. Juan ate an egg with 78 calories and some cereal with 110 calories per serving for breakfast. What is the total amount of calories he consumed? D. A bottle contained 2,000 mL of liquid and is being poured out at an average rate of 300 mL per second. How much liquid is left in the bottle after x seconds?"