Consider an object moving in the plane whose location at time t seconds is given by the parametric equations: x(t)=5cos(at) y(t)=3sin(t). Assume the distance units in the plane are meters. (a) The object is moving around an ellipse with equation: +2 +=1 where a= 5 and b= 3 (b) The location of the object at time t=1/3 seconds is (2.5 2.598 (c) The horizontal velocity of the object at time t is x ' (t)= —5nsin(t) m/s. (d) The horizontal velocity of the object at time t=1/3 seconds is (e) The vertical velocity of the object at time t is y' (t)= 3π cos(л) m/s. m/s. (f) The vertical velocity of the object at time t=1/3 seconds is 4.712 (g) The slope of the tangent line at time t=1/3 seconds is -0.347 m/s. (h) Recall, the speed of the object at time t is given by the equation: s(t)=√√ [x '(t)]² + [y ' (t)]² m/s. The speed of the object at time t=1/3 seconds is (i) The first time when the horizontal and vertical velocities are equal is time t= 0.828 (j) Let Q be the position of the object at the time you found in part (i). The slope of the tangent line to the ellipse at Q is This question builds on an earlier problem. The randomized numbers may have changed, but have your work for the previous problem available to help with this one. A 10-centimeter rod is attached at one end to a point A rotating counterclockwise on a wheel of radius 5 cm. The other end B is free to move back and forth along a horizontal bar that goes through the center of the wheel. At time t=0 the rod is situated as in the diagram at the left below. The wheel rotates counterclockwise at 1.5 rev/sec. At some point, the rod will be tangent to the circle as shown in the third picture. B A B A B at some instant, the piston will be tangent to the circle (a) Express the x and y coordinates of point A as functions of t: x= 5 cos(3) and y= 5 sin(3πt) (b) Write a formula for the slope of the tangent line to the circle at the point A at time t seconds: -cot (3πt) (c) Express the x-coordinate of the right end of the rod at point B as a function of t: 5 cos(3лt) + 10 sin(3лt) (d) Express the slope of the rod as a function of t: -cot (3πt) (c) Find the first time when the rod is tangent to the circle: seconds. (d) At the time in (c), what is the slope of the rod? (e) Find the second time when the rod is tangent to the circle: seconds. (f) At the time in (e), what is the slope of the rod?

The concentration of carbon dioxide (CO2) in the atmosphere is increasing rapidly due to our use of fossil fuels. Because plants use CO2 to fuel photosynthesis, more CO, may cause trees and other plants to grow faster. An elaborate apparatus allows researchers to pipe extra CO, to a 30-meter circle of forest. They selected two nearby circles in each of three parts of a pine forest and randomly chose one of each pair to receive extra CO,. The response variable is the mean increase in base area for 30 to 40 trees in a circle during a growing season. We measure this in percent increase per year. Here are one year's data. Control plot - Control Treated plot 10.547 Pair Treated 9.732 0.815 7.273 9.244 1.971 5.762 8.675 2.913 (a) State the null and alternative hypotheses. (Where u is the mean difference, treated minus control.) Но: и На: и Explain clearly why the investigators used a one-sided alternative. The researchers have no preconceived idea what extra CO, will do to the growth rate.…

The concentration of carbon dioxide (COy) in the atmosphere is increasing rapidly due to our use of fossil fuels. Because plants use COz to fuel photosynthesis, more COz may cause trees and other plants to grow faster. An elaborate apparatus allows researchers to pipe extra CO, to a 30-meter circle of forest. They selected two nearby circles in each of three parts of a pine forest and randomly chose one of each pair to receive extra COz. The response variable is the mean increase in base area for 30 to 40 trees in a circle during a growing season. We measure this in percentage increase per year. Here are one year's data Pair 1 2 3 Control Plot 9.752 7.263 5.742 Treated Plot 10.587 9.244 8.675 Treated - Control 0.835 1.981 2.933 State the null and alternative hypotheses. Explain clearly why the investigators used a onesided alternative. Use at procedure to carry out a test and report your conclusion in simple language. The investigators used the test you just carried out.Any use of the…

You work as a data scientist for a real estate company in a seaside resort town. Your boss has asked you to discover if it's possible to predict how much a home's distance from the water affects its selling price. You are going to collect a random sample of 9 recently sold homes in your town. You will note the distance each hame is from the water (denoted by X, in km) and each home's selling price (denoted by y, in hundreds of thousands of dollars). You will also note the product X-y oaf the distance from the water and selling price for each hame. (These products are written in the row labeled "Xy")- (a) Click an "Take Sample" to see the results for your random sample. Distance from the water, X 3.8 21 1.3 0.2 29 21 4.4 1.3 1.8 (In lam) Selling price, y (In hundreds of thousands of dollars) Tako Sample 5.7 10.3 17.1 14.8 8.2 1.4 6.1 10.4 9.6 xy 21.66 21.63 22.23 2.96 23.78 15.54 26.84 13.52 17.28 Sand data to calculator v Based on the data from your sample, enter the indicated values…