It is your friend's birthday tomorrow, and you are planning a surprise party for them. For this party, you got perfectly spherical balloons and you are filling these When studied closely, biologists found that population growth does not only balloons with helium. The helium enters that balloon at a rate of 120 cm /s (i.e. depend on the size of the population, but also the density of the population. if an the volume of the balloon is increasing at this rate). How quickly is the radius of area is too densely populated, there will be too much competition for food, mates, the balloon changing when the diameter of the balloon is 60 cin? Do not include and nesting sites, leading to a decline in numbers. This type of growth is known as units in your answer. density dependent growth. A common example of this type of growth is known as logistic growth, which can be represented by the growth rate G(N): G(N) dr di m/s where N is the population size (the independent variable), r is a constant called the intrinsic growth rate, and K is the carrying capacity of the environment, also a constant. Find the population size N* at which the g rate is maximized for logistic growth You wish to create a very luxurious can (ie. cylinder) without a top that can hold 2 volume of 425 mL. The cost of the material to make the can is however quite expensive. Assuming that the cost of the material is proportional to the amount of material needed to build the can, find the dimensions of the can such that the cost of the material is minimized. Submit your answer in numerical form, and round to Consider a police car sounding a siren with a frequency fo while traveling right two decimal places. towards you with velocity u. Because of the Doppler effect, the frequency f that 3 you measure is not the same as the original frequency fo of the siren. More Radius: cm precisely, the two are related by Height: cm S = where u, is the speed of sound (which you can take to be u, = 343 m/s, which is approximately correct in dry air at 20" C ), and u is the velocity of the car as measured in a coordinate system centred on the observer with positive direction pointing towards the approaching car (and hence a car approaching the observer will have negative velocity). Suppose that the original frequency of the siren is fo = 1280 Hz. At a certain time, you measure the frequency of the siren to be f = 1330 Hz, and increasing at a rate of 5 Hz/s. What is the police car's velocity (in m/s) and acceleration (in m/s?) at the time of observation? (Enter your answers in exact form.) The velocity is u= m/s The acceleration is a = m/s 4 Are these numbers realistic