Derivatives : Maximum and Mini

1). An open rectangular box with square ends is to hold 6400 cm^3. It is built at a cost of $ 75/ cm^2 for the base and $25/ cm ^2 for the sides. Find the most economical dimensions.

2). A wall 8 meter high is 3 3/8 meter from a house. Find the shortest ladder which will reach from the ground to the house when leaning over the wall.

3). A company has the following schedule of charges : $30 per 1000 for orders of 50 000 or fewer , with the charge per thousand decreased by $ 0.375 for each thousand above 50 000. Find the order that will make the receipts a maximum.

4). Find the minimum distance from the point ( 3 , 4 ) to the parabola
Y^2 = 8x

Derivatives : Maximum and Mini

9). Find the dimensions of a right circular cone of minimum volume that can be circumscribed about a sphere of radius 8cm.

32 , 8root

11. The sum of 2 positive numbers is 20. Find the numbers if
A) their product is a maximum
B) the sum of their squares is a minimum
C) the product of the square root of one and the cube root of the other is a maximum

12). The product of 2 positive numbers is 16. Find the numbers if
A) their sum is a minimum
B) the sum of one and the square of the other is a minimum

Derivatives : Maximum and Mini

5). The total cost of producing x radio sets per day is $ ( 1/4 x^2 + 35x + 25 ) and the price per set is at which they may be sold is $ ( 50 – 1/2x ). Find the daily output for maximum profit.

Answer 10 sets /day

6). The cost of fuel in running a locomotive is proportional to the square of the speed and is $25/hr for a speed of 25km/hr. The other costs are $100 /hr regardless of speed. Find the speed that will make the cost per kilometer a minimum.

Answer 50km/hr

8) A rectangular field , to enclose a given area , is fenced off along a straight river. If no fencing is needed along the river , show that the least amount of fencing will be required when the length of the field is twice the width ( this is all the info available- the condition ? length =2x width).