(84ftx112ft)3.Find the volume of the largest box that can be made by cutting equal squares out of the corners of a piece of cardboard of dimensions 15 inches by 24 inches, and then turning up the sides. (486)4.Find the rectangle of maximum perimeter inscribed in a given circle. 5.A page is to contain 24 sq. in. of print. The margins at top and bottom are 1.5 in., at the sides 1 in. Find the most economical dimensions of the page. (6x9)6.A ship lies 6 miles from shore, and opposite a point 10 miles farther along the shore another ship lies 18 miles offshore. A boat from the first ship is to land a passenger and then proceed to the other ship. What is the least distance the boat can travel? (26 mi)7.Find the shortest distance from the point (5, 0) to the curve 2y2= x3. (√𝟏𝟑)8.A cylindrical tin boiler, open at the top, has a copper bottom. If sheet copper is m times as expensive as tin, per unit area, find the most economical proportions. 9.A man on an island 12 miles south of a straight beach wishes to reach a point on shore 20 miles east. If a motorboat, making 20 miles per hour, can be hired at the rate of $2.00 per hour for the time it is actually used, and the cost of land transportation is $0.06 per mile, how much must he pay for the trip? (2.16)10.A right circular cylinder of radius r and height h is inscribed in a right circular cone of radius 6 m and height 12 m. Determine the radius of the cylinder such that its volume is a maximum.

SPTOPICS ALGEBRA PREPARED BY: ENGR. LIONEL P. LAPUZ TIME RATES If a quantity x is a function of time t, the time rate of change of x is given by dx/dt. When two or more quantities, all functions of t, are related by an equation, the relation between their rates of change may be obtained by differentiating both sides of the equation with respect to t. Steps in Solving Time Rates Problem Identify what are changing and what are fixed. Assign variables to those that are changing and appropriate value (constant) to those that are fixed. Create an equation relating all the variables and constants in Step 2. Differentiate the equation with respect to time. SAMPLE PROBLEMS 1.Water is flowing into a vertical cylindrical tank at the rate of 24 ft^3/min. If the radius of the tank is 4 ft, how fast is the surface rising? (0.477 ft/min)2.A rectangular trough is 10 ft long and 3 ft wide. Find how fast the surface rises, if water flows in at the rate of 12 ft^3/min. (0.4 ft/min)3.A ladder 20 ft long leans against a vertical wall. If the top slides downward at the rate of 2 ft/sec, find how fast the lower end is moving when it is 16 ft from the wall. (1.5 ft/s)4.A train starting at noon, travels north at 40 miles per hour. Another train starting from the same point at 2 PM travels east at 50 miles per hour. Find, to the nearest mile per hour, how fast the two trains are separating at 3 PM. (56.15 mi/hr)5.A man 6 ft tall walks away from a lamp post 16 ft high at the rate of 5 miles per hour. How fast does the end of his shadow move? (8mi/hr)6.A trapezoidal trough is 10 ft long, 4 ft wide at the top, 2 ft wide at the bottom and 2 ft deep. If water flows in at 10 ft^3/min, find how fast the surface is rising, when the water is 6 in deep. (0.4 ft/min)