If anyone could help me with these questions, please do so! Also please provide detailed and correct solutions on paper. Thank you so much!

WK Exit Fullscreen la) Sketch a possible distance curve for the velocity curve at right. Justify your solution by offering a full explanation of your reasoning. 1b) Sketch a possible distance curve for the acceleration curve at right. Justify your solution by offering a full explanation of your reasoning. 2) A baseball is hit vertically upward. The position function s(t), in metres, of the ball above the ground is s(t) = -5t2 + 30t + 1 , where t is in seconds. a) Determine the maximum height reached by the ball. b) Determine the velocity of the ball when it is caught 1 m above the ground. 3) An object is moving horizontally. The object's displacement, s, in metres at t seconds is described by the function s(t) = 4t - 7t2 + 213 a) Determine the velocity and acceleration at t = 2 seconds. b) When is the object stationary? (ie. find the time, t, when the velocity is zero (0)). c) At what time, to the nearest tenth of a second, is the acceleration equal to 0? 4. Determine the first and second derivative of the following functions. a) h(x) = 3x4 - 4x3 - 3x2 - 5 b) f (x) = (2x - 5)3 cy = I + 31. A 1000 L tank is draining such that the volume V of water remaining in the tank after t minutes is V = 1000/ 1- Find the rate at which the water is flowing out of the tank after 8 min. 2. When a certain object is placed in an oven at 540'C, its temperature T(t) rises according to the equation T(t) = 540(1 -e .!), where t is the elapsed time (in minutes). What is the temperature after 9 minutes and how quickly is it rising at this time? 3. The amount of daylight a particular location on Earth receives on a given day of the year can be modelled by a sinusoidal function. The amount of daylight that Sarnia, Ontario will experience in 2018 can be modelled by the function - D(t) = 12.18 + 3.1 sin(0.017t - 1.376) , where t is the number of days since the start of the year. a. On January 1, how many hours of daylight does Sarnia receive? b. What would the slope of this curve represent? c. The summer solstice is the day on which the maximum amount of daylight will occur. On what day of the year would this occur? (Check this using Internet research (ie. Google search on "date of summer solstice"). d. Verify this fact using the derivative. e. What is the maximum amount of daylight Sarnia receives? f. What is the least amount of daylight Sarnia receives? 4. The following table indicates a number of households (in thousands) with a total income under $20,000 or over $100,000. 2014 2015 2017 2018 2019 Under $20,000 625.03 591.76 595.05 586.30 566.98 Over $100,000 1,248.48 1,409.19 1,538.54 1,635.93 1,803.71 a. Use Desmos, Excel or Curve Expert to help you model each of the two income segments with an appropriate function. (state the polynomial equation that models this data and yields a strong r - value). b. Which segment of the population is changing more quickly in 2015? c. Are the results in this table sufficient to show that poverty is decreasing? What additional information would you like to know in order to make your conclusions?Creating Models + + + + 1. A window has the shape of a rectangle, topped by a semicircle, as shown. The perimeter of the window is 8 m. Determine an equation that could be used to express the area of the window in terms of its width. Express this as a function, A(w). You need not solve the problem. O 1 km 2. An oil pipeline will connect points A and B which are 3 km apart and on opposite banks of a 1 km wide river. From point A to point C on the opposite bank, the pipeline will run under the water. From point C to point B the pipeline will run above ground. The cost of running the pipeline under water is three times the cost of running it above ground. kn Determine an equation that can be used to minimize the cost. You need not solve the problem. BDetermining Boundaries 3. A wire 60 cm long is to be cut into two pieces. One of the pieces will be bent into the shape of a square and the other into the shape of an equilateral triangle, as shown in the diagram below: 60 cm x cm 60 - xcm 60-X cm - cm 3 The wire is to be cut in order that the sum of the areas of the square and the triangle is to be a maximum. An equation that can be used to model the sum of the areas is A (a) = 16 + V3(60-x)2 36 . Determine the boundaries and the corresponding areas. You need not solve further. 4. A cylindrical tank is to have a capacity of 1000 m3. It is to fit into a foundry that is 12 m wide with a height of 11 m. The base of the tank will cost half as much as the top. The metal for the side of the tank will cost four fifths as much as the top. An equation that can be used to model the cost of the tank is C (r) = ;mr +- 1600 r Determine any restrictions on r. You need not solve further. Optimization 5. A bus service carries 10 000 people daily between Ajax and Union Station. The bus company has space to serve up to 15 000 people each day. The cost to ride the bus is $20/day. Market research shows that if the fare is increased by $0.50/day, then 200 fewer people will ride the bus. What fare should be charged to get a maximum revenue, given that the bus company must have at least $130 000 in fares a day to cover operating expenses?6. A farmer wants to fence a rectangular area of 800,000 ma and divide it in half with a fence parallel to one of the sides of the rectangle. The middle fences is twice as expensive, so doubled (ie just put two fences down the middle). How can this be done in order to minimize the perimeter and thus, minimize the cost of the fence? (ie. Find the dimensions of the rectangular area that use the minimum amount of fencing). 7. A crocodile is stalking it's prey located 20 metres upstream on the opposite bank of a river. Crocodiles travel at different speeds on 20 metres land and in the water. The time taken for the x metres crocodile to reach it's prey can be minimised if it swims to a particular point P, x metres upstream on the opposite side of the river as shown in the diagram. The time taken, T , measured in tenths of a second is given by the formula T(x) = 5\\(36 + x2) + 4 (20 -x) a) Calculate the time taken if the crocodile does not travel on land ? b) Calculate the time taken if the crocodile swims the shortest distance possible ? c) Between these two extreme there is one value of x which minimizes the time taken. Find this value of x and hence calculate the minimum possible time. 8. Sophie is given a piece of sheet metal that is twice as long as it wide. It has an area of 800m- . She needs to find the dimensions of the rectangular box that would contain a maximum volume if it were constructed from this piece of metal by cutting out squares of equal area at all four corners and folding up the sides. The box will not have a lid. Give your answer correct to one decimal place.1. Given the function M(t) = 215 - 31- - 36t, find the critical values and determine, using both the second derivative test and a sign chart, the nature of these critical values. 2. A projectile is launched with a velocity of 27 m/s at 55 to the ground. Determine its horizontal and vertical velocities. ... . .... .. .. . . . . . .. 55 3. Two trains start from the same point at the same time, one going east at a rate of 40 km/h and the other going south at 60 km/h, as shown in the diagram at right. Find the rate at which they are separating after 1 h of travel. Train A 40 km/h Train B 60 km/h Distance Between .....................= Trains4. A professional basketball team plays in a stadium that holds 23.000 spectators. With ticket prices at $60. the average attendance had been 13000. When ticket prices were lowered to $55. the average attendance rose to 20.000. Based on this pattern. how should ticket prices be set to maximize ticket revenue? 5. Higher-order derivatives are derivatives that are taken found after a previous derivative has been taken. For example, we can lake the first derivative (y') and the second derivative (y") and the 3rd derivative {y"') it we are given the original function (y). Quite often. multiple higherrorder derivatives can be found. Higherrorder derivatives are used in Science (physics) and mathematics (Differential Equations). A differential equation is an equation that contains functions and some higherorder derivatives. Using a formal proof structure (LHS : RHS). prove the dil'lerential equation xy' - 3y 4- y\" 4- A = 2x2 given the function y = x5 - 2:2 + 3x. 6. A 5.000 m3 rectangular area of a eld is to be enclosed by a lence. with a moveable inner fence built across the narrow part of the field, as shownJ'he perimeter fence costs $10t'm and the inner fence costs Wm. Determine the dimensions of the field to minimize the cost. ?. The foilowing table displays the historical number of HIV diagnoses per year in a particular country. 199? 1993 1999 2000 2001 2002 2003 2004 2005 251.2 2343 2230 2113 21.?3 2495 2496 2533 2513 a. Using Desmos. Excel or Curve Expert or another curve modelling program, determine an equation that can be used to model this data. b. Using this model. estimate the number of diagnoses in 1996 and in 2006. c. At what rate would the number of diagnoses be changing in 2006? d. Halfway through 2006. the number of new HIV diagnoses was found to be 1232. Assuming this rate stays fairly constant for the remainder of the year. does this new information change the modelling equation? If so. how would this change your answer to part (c)? If you were an advocate for furthering HIV and AIDS research and treatment programs. would you be encouraged or discouraged by these results