5 1. The sum of a number and the square of another number is 27. Find the numbers so that their product is a maximum. 2. A rectangular box 1with an open top has a square base. The volume of the box is 103 cm". If the box is made of the least amount of material, what must be its dimensions? 6 3. A farmer wants to build two adjacent and identical rectangular cattle pens on 60 000 m of land. What is the least amount of fencing material required?5 4. A children's picture book is being designed so that each page contains 320 cm of print and pictures, surrounded completely by a white border. The border must be 2 cm wide at the bottom and on each side and 3 cm wide at the top. Calculate the dimensions of the page of smallest possible area. Note: Verification that the dimensions yield a minimum is not needed for this question.5 5. Sarah is in a kayak 250 m offshore from point A on a straight beach. A storm is brewing and Sarah wishes to go to a shelter located 900 m down the beach from point A. If Sarah can paddle her kayak at 2 km/h and walk at 3 km/h, how far down the shore should she beach her kayak to reach the shelter in the shortest amount of time? Note: Verification that the time is a minimum is not needed for this question. 6. A property management company manages an apartment block containing 15E! units. All 150 units are rented at a monthly rate of 54s] per unit and each unit costs the property management company $11.30 Imonth for utilities and repairs. For every 325 rent increase, four fewer apartments are occupied. What rent should be charged in order to realize the most profit? 7. A spherical balloon is being inflated. 2 a. Find the rate of change of the volume when the radius is - cm and changing at a rate of 2 cm/min. 3 b. If the volume of the balloon is increasing at a rate of 10 cm /min, how fast is the diameter increasing when the radius is 3 cm?3 8. A block of ice has a square top and bottom and rectangular sides. At a certain point in time, the square top and bottom each have a length of 30 cm, which are decreasing at a rate of 2 cm/h. At the same time, the height of the ice block is 20 cm and decreasing at 3 cm/h. How fast is the the volume of the ice block decreasing?3 9. The angle of elevation of the sun is decreasing at 7 rad/h. How fast is the shadow cast by a tree 10 m tall lengthening when the angle of elevation of the sun is rad?A student leaves school on his skateboard heading east. He travels at a rate of 13 kmt. At the same time. another student on a bike, leaves the school heading directly south at a rate of 24 krnih. At what rate is the distance between the two students increasing when the skateboarder is 45 km from the school and the biker is 3 km from the school? 3 11. A water tank is built in the shape of a circular cone with a height of 5 m and a diameter of 6 m at the top. Water is pumped into the tank at a rate of m /min. Find the rate at which the water level is rising when the water is 2 m deep