Part 30

9. Find the geometrical progression whose sum to infinity is 4, and whose second term is 3/4.

10. Solve 4x + 4[3x^2 - 7x + 3]^(1/2) = 3x^2 - 3x + 6.

11. Solve { 2x^2 + 3xy - 5y^2 = 4, { 2xy + 3y^2 = -3.

12. Two hundred stones are placed on the ground 3 feet apart, the first being 3 feet from a basket. If the basket and all the stones are in a straight line, how far does a person travel who starts from the basket and brings the stones to it one by one?

Solve graphically; and check by solving algebraically:

1. { x^2 + y^2 = 25, { x + y = 1.

2. x^2 - 3x - 18 = 0.

3. x^2 + 3x - 10 = 0.

Determine the value of m for which the roots of the equation will be equal: (HINT: See page 40. To have the roots equal, b^2 - 4ac must equal 0.)

4. 2x^2 - mx + 12-1/2 = 0.

5. (m - 1)x^2 + mx + 2m - 3 = 0.

6. If 2a + 3b is a root of x^2 - 6bx - 4a^2 + 9b^2 = 0, find the other root without solving the equation. (Univ. of Penn.)

7. How many times does a common clock strike in 12 hours?

8. Find the sum to infinity of 2/(2^(1/2)), 1/(2^(1/2)), 1/(2^(1/2)), ....

9. Solve [x/2 + 6/x]^2 - 6[x/2 + 6/x] + 8 = 0.

10. Find the value of the recurring decimal 2.214214....

11. A man purchases a $500 piano by paying monthly installments of $10 and interest on the debt. If the yearly rate is 6%, what is the total amount of interest?

12. The arithmetical mean between two numbers is 42-1/2, and their geometrical mean is 42. Find the numbers. (College Entrance Exam. Board.)

13. If the middle term of [3x - (1)/(2[x^(1/2)])]^4 is equal to the fourth term of [2[x^(1/2)] + 1/2x]^7, find the value of x. (M. I. T.)

~PROBLEMS~

~Linear Equations, One Unknown~

1. A train running 30 miles an hour requires 21 minutes longer to go a certain distance than does a train running 36 miles an hour. How great is the distance? (Cornell.)

2. A man can walk 2-1/2 miles an hour up hill and 3-1/2 miles an hour down hill. He walks 56 miles in 20 hours on a road no part of which is level. How much of it is up hill? (Yale.)

3. A physician having 100 cubic centimeters of a 6% solution of a certain medicine wishes to dilute it to a 3-1/2% solution. How much water must he add? (A 6% solution contains 6% of medicine and 94% of water.) (Case.)

4. A clerk earned $504 in a certain number of months. His salary was increased 25%, and he then earned $450 in two months less time than it had previously taken him to earn $504. What was his original salary per month? (College Entrance Board.)

5. A person who possesses $15,000 employs a part of the money in building a house. He invests one third of the money which remains at 6%, and the other two thirds at 9%, and from these investments he obtains an annual income of $500. What was the cost of the house? (M. I. T.)

6. Two travelers have together 400 pounds of baggage. One pays $1.20 and the other $1.80 for excess above the weight carried free. If all had belonged to one person, he would have had to pay $4.50. How much baggage is allowed to go free? (Yale.)

7. A man who can row 4-1/3 miles an hour in still water rows downstream and returns. The rate of the current is 2-1/4 miles per hour, and the time required for the trip is 13 hours. How many hours does he require to return?