« ΠροηγούμενηΣυνέχεια »
4. 4(3 b— x)=3(2b+x).
w 2 x+3 4x+2 1
10. * +=2
II li colo ole
2 x+k r
25. a(b+x) – (6+x)(a − x)=x2+
107. Formulas. If a person travels for 10 hours at the rate of 15 miles an hour, we know from arithmetic that the total distance he goes will be 15X10=150 miles. Stated in general (algebraic) language, we can say in the same way that if a person travels t hours at the rate of r miles an hour, the distance s which he will go is given by the formula (called the law of uniform motion)
s=rt. This is a literal equation which expresses the value of s in terms of r and t. If we wish, we can solve this equation for t, giving t= », and what we now have is t expressed in terms of s and r. Or, we can solve the original equation for r, which gives r=, and this expresses r in terms of s and t.
This illustrates the very important fact that in nearly all branches of knowledge, especially in engineering, geometry, physics, and the like, there are general laws which express themselves in mathematical formulas. Such formulas are really nothing but literal equations in which two or more letters appear, and it is often desirable to solve such equations for some one letter in order to find its value in terms of the others.
EXERCISES — APPLIED PROBLEMS 1. The area A of a rectangle whose dimensions (length and breadth) are a and b is given by the formula Arab. Solve this for a. Also, solve for b. In each case state what letters your answer is in terms of.
2. The formula for the area A of a triangle whose height (altitude) is h and whose base is a is A=ļah.
Solve for a and solve for h, and state in each case what letters your answer is in terms of.
3. Solve for r in the formula C =2 or. (This is the formula for the circumference of a circle in terms of its radius. See Ex. 22, p. 22.)
4. Solve for B in the formula A=ih(B+b). (This is the formula for the area of a trapezoid whose bases are B and b and whose altitude is h. See Ex. 28, p. 88.)
5. The interest I which a principal of p dollars will yield in t years at r per cent is determined by the formula I = prt/100. Solve this for r and use your result to answer the following question: What rate of interest is necessary in order that $50 may yield $6 interest in two years' time? Solution. Solving for r gives
pt We now have only to see what this becomes when we put I=6, p=40, and t=2. The result is
-- 100X6=6%. Ans.
50X2 6. Using the interest formula of Ex. 5, solve it for p and use your result to answer the following question: How great a principal must be invested at 5% in order that it will have yielded $90 in interest by the end of three years ?
7. It is shown in solid geometry that the volume V of any pyramid is equal to one third the product of the area B of its base multiplied by the height h. That is, we have the formula V= Bh. Solve this for h and use your answer to find how high
a pyramid must be to contain one Fig. 40.
cubic foot, provided its base contains 1 square foot. Answer the same when the base contains only 12 square inches.
8. The formula for converting degrees Fahrenheit to degrees Centigrade is
C= (F-32). Solve for F and use your result to answer the following questions:
(a) How many degrees Fahrenheit correspond to 0° Centigrade?
(6) How many degrees Fahrenheit correspond to 100° Centigrade?
9. The figure represents a body (of any material) submerged in a liquid. If the base of the body contains A square feet and is everywhere at a depth of h feet from the top of the liquid, then the upward pressure P on the base (due to the liquid) is given by the formula P=wAh, where w represents the weight of 1 cubic
Fig. 41. foot of the liquid.
Find the pressure per square foot of surface near the bottom of a standpipe in which the water is 40 feet high, it being given that fresh water weighs 62.5 lb. per cubic foot.
10. Find by the formula in Ex. 9 the pressure per square foot at the bottom of the ocean at a depth of 3000 feet, it being given that sea water weighs 64 pounds per cubic foot.
11. The bottom of a rectangular cistern is 6 feet square. For what depth of water will the total pressure on the bottom be 18 tons ? [Hint. Solve the formula of Ex. 9 for h.]
12. How deep in the ocean can a diver go without danger in a diving armor that can safely sustain a pressure of no more than 140 pounds per square inch?
RATIO AND PROPORTION 108. Ratio. The quotient of one number divided by another of the same kind is called their ratio.
Thus, the ratio of 12 inches to 6 inches is the fraction *, or 4. The ratio of 2 feet to 3 feet is the fractions. The ratio of 10 cents to $1 is 4%, or to. Note that in every case a ratio is simply a fraction of the kind studied in arithmetic.
The first number, or dividend, is called the antecedent; the second number, or divisor, is called the consequent.
Thus, in the ratio , the antecedent is 6 and the consequent is 7.
EXERCISES 1. What is the ratio of 5 quarts to 8 quarts ? of 5 quarts to 10 quarts ?
2. What is the ratio of 18 inches to 3 inches? of 18 inches to 1 foot ?
3. What is the ratio of a foot to a yard? of a yard to an inch?
4. A stick was divided into two parts one of which contained 2 units and the other 7 units. What was the ratio of the two parts ?
5. State (as a fraction in simplest form) the value of each of the following ratios.
(a) 8 to 12. (c) 72 to 36. (e) 30 to 10. (g) 60 to 15. (6) 15 to 8. (d) 80 to 75. (f) 18 to 24. (h) 7.5 to 2.5