1) A cylinder of radius xx is made to fit inside a sphere of radius R as shown below. (I) show that the volume of the cylinder is given by V=2πh(R²-h²)
(ii) if the radius of the sphere, R, is 12cm, find the maximum volume of the cylinder.
(iii) use the second derivative test to show that your solution gives a maximum.

2) A metal box (without a top) is to be constructed from a square sheet of metal that is 20 cm
on a side by cutting square pieces of the same size from the corners of the sheet and then
folding up the sides. Find the dimensions of the box with the largest volume that can be
constructed in this manner and prove that these values give a maximum volume.

1. A spherical balloon is deflating so that its volume is decreasing at a constant rate of 12
cm3 per hour.

Find the rate at which the surface area of the balloon is decreasing when the volume of
the balloon is 2 500cm3.
V=4/3πr³
S=4πr²
Show any derivatives that you need to find when solving this problem.

2. A cylindrical piece of wood is being turned on a lathe. If it is 20cm long and its
radius is decreasing at a rate of 2cm per minute, find the rate at which the
volume is decreasing when its diameter is 5cm.
V = πr²h

The upcoming championship high school football game is a big deal in your little town. the problem is, it is being played in the next biggest town, which is two hours away! to get as many people as you can to attend the game, you decide to come up with a ride-sharing app, but you want to be sure it will be used before you put all the time in to creating it. you determine that if more than three students share a ride, on average, you will create the app. you conduct simple random sampling of 20 students in a school with a population of 300 students to determine how many students are in each ride-share (carpool) on the way to school every day to get a good idea of who would use the app. the following data are collected: 6 5 5 5 3 2 3 6 2 2 5 4 3 3 4 2 5 3 4 5 construct a 95% confidence interval for the mean number of students who share a ride to school, and interpret the results. part a: state the parameter and check the conditions. part b: construct the confidence interval. be sure to show all your work, including the degrees of freedom, critical value, sample statistics, and an explanation of your process. part c: interpret the meaning of the confidence interval. part d: use your findings to explain whether you should develop the ride-share app for the football game.