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Two posts, one 8 feet high and the other 12 feet high, stand 15 feet apart. They are to be stayed by wires attached to a single stake at ground level, the wires running to the top of the posts. Where the stake should be placed, to use the least amount of wire?
A manufacturer makes aluminum cups of a given volume (16 cubic inches) in the form of right circular cylinders open at the top. Find the dimensions which use the least material.
Find the dimensions of the largest circle that can be inscribed in a square of 12 inches.
A closed box with a square base is to have a volume of 2000 cu inches. The material for the top and bottom of the box is to cost 3 pesos per square inch, and the material for the sides is to cost 1.50 pesos/square inch. If the cost of the material is to be least, find the dimensions of the box.
A right triangle has hypotenuse of length 13 and one leg of length 5. Find the dimensions of the rectangle of largest area which has one side along the hypotenuse and the ends of the opposite side on the legs of this triangle?
A box with square base is to have an open top. The area of the material in the box is to be 100 in square. What should the dimensions be in order to make the volume as large as possible?
Suppose that you need to deliver the message “161803398” which is the pass key for a weapon
activation to country X. Encrypt the message using Caesar cipher with the encryption key,
2
( 2 )mod9 n
where
n 1, 2,...,9 , without being intercepted and decrypted by other countries.
(a) State the decrypted message.
(b) Determine the decryption key.
(c) Suggest an improvement to the encryption key to increase the encryption strength
activation to country X. Encrypt the message using Caesar cipher with the encryption key,
2
( 2 )mod9 n
where
n 1, 2,...,9 , without being intercepted and decrypted by other countries.
(a) State the decrypted message.
(b) Determine the decryption key.
(c) Suggest an improvement to the encryption key to increase the encryption strength
Find the numbers whose sum is x , if the product of one by the square of the other is to be a maximum.
A particle moves along a straight line according to the law : S = 132 + 10t - 6t^2 + 3t^3.
Find the following:
a. The velocity and acceleration at any time t?
b. Velocity at t = 2;
c. Acceleration at t = 3.
Find the following:
a. The velocity and acceleration at any time t?
b. Velocity at t = 2;
c. Acceleration at t = 3.
Find an equation of each normal line to the curve: y = x^3 - 4x that is parallel to the line
x + 8y - 8 = 0
x + 8y - 8 = 0
