For a concert, a circular stage is constructed and placed in the center so fans can completely surround the musicians. To record the sound of the crowd, two directional microphones are placed next to each other on the stage, one facing due east and the other facing due west. The patterns of the microphones can be represented by the polar equations $r=2.5+2.5 \cos \theta$ and $r=-2.5-2.5 \cos \theta.$ Sketch the graph of each microphone pattern on the same polar grid.

In many U.S. parks, campers must secure food and provisions from bears and other animals. One method is to secure food using a bear bag, which is done by tossing a bag tied to a rope over a tall tree branch and securing the rope to the tree. Suppose a tree branch is 30 feet above the ground, and a person 20 feet from the branch throws the bag from 5 feet above the ground. Will a bag thrown at a speed of 45 feet per second at an angle of $75^{\circ}$ go over the branch?

Write an equation for a circle that satisfies each set of conditions. Then graph the circle. center at (-4,-3), tangent to y=3

Consider f(x) = x + 3 and g(x) = cos x, where x is in radians. a) Describe and sketch a graph of f(x). Is this function even, odd, or neither? b) Describe and sketch a graph of g(x). Is this function even, odd, or neither? c) Predict the shape of y = f(x)g(x). Sketch a graph of your prediction. d) Check your work by graphing the three functions in parts a) to c) using graphing technology. e) Give the domain and the range of y = f(x)g(x).

Draw sketches of possible graphs for which the following data hold. Consider only the domain $0 \leqslant x \leqslant 5$, and assume that the graph of $y=f^{\prime \prime}(x)$ is smooth. (a) f(0) = 0, f(2) = 5, f'(2) = 0, f"(2) < 0, f(4) = 3, f'(4) = 0, f"(4) > 0 (b) f(0) = 0, f"(1) < 0, f(2) = 5, f'(2) = 0, f"(3) > 0, f(4) = 7 (c) f(0) = 0, f'(0) = -1, f'(1) = 0, f(3) = 0, f'(3) = 2, f"(3) = 0, f"(4) < 0 (d) f(0) = 1, f'(0) = 1, f''(0) = 1 and f"(x) increases as x increases (e) f(0) = 1, f'(0) = 0, f'(x) < 0 for 0 < x < 5, f(5) = f'(5) = 0 (f) f(0) = 3, f'(0) = -2, f"(x) > 0 for 0 < x < 5, f(5) = f'(5) = 0

Anthony measured the dimensions of a rectangular box to be 45 cm by 35 cm by 2 m. Determine the volume of the box in cubic meters.

Find the maximum and minimum values of the objective function f(x, y) and for what values of x and y they occur, subject to the given constraints.

$$\begin{aligned} &f(x, y)=3 x+6 y\\ &y \leq-\frac{1}{2} x+\frac{5}{2}\\ &y \leq 2, y \geq 0\\ &x \leq 3, x \geq 0 \end{aligned}$$

The factory will create a lot of noise and fumes. The area is one with a high level of unemployment - people's incomes are below the national average. Before deciding whether or not to grant planning permission the government also considers the external costs and external benefits - that is, the impact on the rest of society other than the business itself.

$$\begin{matrix} \text{External costs} & \text{External benefits}\\ \text{Waste products will cause pollution} & \text{Jobs will be created}\\ \text{Smoke and fumes may damage the health of residents} & \text{Other firms may move into the area to provide services to the chemical firm}\\ \text{Parkland cannot now be used by local residents} & \text{The chemical factory will pay taxes - government might increase spending on social projects e.g. more hospitals}\\ \end{matrix}$$

The government will try to give a value to all of these costs and benefits. This is called cost-benefit analysis. This is not easy to do. For example, what is the cost of losing parkland for children to play on? Estimates are made and then the total costs and benefits of the decision are added up: private costs, added to external costs, give a total social cost, private benefits, added to external benefits, give a total social benefit figure. If the total social benefit is greater than the total social cost, the scheme is likely to be accepted. If, however, the total social cost is greater than the total social benefit, the government will probably refuse permission. Explain why any two stakeholder groups will be worried about the external costs from a new chemical factory.

Find a value of a so that each objective function has maximum values at the indicated vertex, subject to the following constraints.

$$\begin{aligned} &\begin{array}{l} {x \geq 0} \\ {y \geq 2} \end{array}\\ &x+y \leq 9\\ &-4 x+3 y \leq 6 \end{aligned}$$

$f(x, y)=-4 x+a y,(3,6)$

(*) At one time, chemists thought that the for mula of a binary compound was the simplest formula possible. For example, the chemical formula of water was thought to be HO and that of ammonia was thought to be NH. Given that 0.832 grams of oxygen combines with 0.104 grams of hydrogen, calculate the atomic mass of oxygen based upon HO as the for mula for water. Similarly, given that 0.403 grams of nitrogen combines with 0.0864 grams of hydrogen, calculate the atomic mass of nitrogen based upon NH as the formula for ammonia.

Many people believe that an Increase in the price of gasoline will change consumer attitudes and driving behavior. They assume people will drive less often and buy smaller, more efficient cars the price of gasoline increases. However, Imagine that gas prices Increased significantly and people's driving habits and gas consumption levels changed very little. Use this information and your knowledge about elasticity of demand to answer. How might time affect this scenario?

Write two goals that you think would benefit society, and describe an incentive that will accomplish each goal. Goal: ________ Incentive: _________

Describe the distribution and any interesting patterns you notice in the stem-and-leaf plot. Interpret your findings in terms of the number of medals won in the 2016 Summer Olympics.

Horsepower is a unit of power; one horsepower (1 hp) is equivalent to 745 watts. The world record for climbing a 25-feet rope was set by Garvin Smith in Los Angeles in 1947, when he climbed the 25 feet in 4.7 seconds. Assuming that Mr. Smith has a mass of 65 kg, calculate the average power (in both watts and horsepower) that he generated during his climb.