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The art of problem solving - answer

At a certain time, Janice notices that her digital watch read a

minutes after two 0' clock. 15 min. later, it reads b min. after

three o' clock. She is amused to note that a is six times greater

than b. What time was it when she looked at her watch for the

second time?

2:54 and 3:09
a+15 = 60+b
as time has changed from 2 to 3 ( 1 hr = 60 minutes)
and we have a = 6b

so use the above two equations we get
5b=45 and b =9
therefore b =09 and a = 9*6 = 54
the times are 2:54 and 3:09

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Math Word Problems-Car Uphill downhill

A car travels downhill at 72 mph (miles per hour), on the level at 63 mph, and uphill at
only 56 mph. The car takes 4 hours to travel from town A to town B. The return trip
takes 4 hours and 40 minutes. Find the distance (in miles) between the two towns

Let the total distance travelled downhill, on the level, and uphill, on the outbound journey, be x, y, and z, respectively.
The time taken to travel a distance s at speed v is s/v.

Hence, for the outbound journey

x/72 + y/63 + z/56 = 4

While for the return journey, which we assume to be along the same roads

x/56 + y/63 + z/72 = 14/3

It may at first seem that we have too little information to solve the puzzle. After all, two equations in three unknowns do not have a unique solution. However, we are not asked for the values of x, y, and z, individually; but for the value of x + y + z.

Multiplying both equations by the least common multiple of denominators 56, 63, and 72, we obtain

7x + 8y + 9z = 4 · 7 · 8 · 9
9x + 8y + 7z = (14/3) · 7 · 8 · 9

Now it is clear that we should add the equations, yielding

16(x + y + z) = (26/3) · 7 · 8 · 9

Therefore x + y + z = 273; the distance between the two towns is 273 miles.

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Euclid's Algorithm

A pair of positive integers (x, y) satisfies the equation 31x + 29 y = 1125. What is x + y? ans - x+y=37

31x+29y=1125

31x+29y=1 we get for x=-14 and y=15

general solution is x =-14*1125+29*t y = 15*1125-31t

14*1125/29 <= t<= 15*1125/31 t is an integer
we get t as 544

plug in t back in the equations above
to get x =26 and y=11
so x+y=37

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Asymptotes

Find the asymptotes of the function

(x^2-4x+5)

f(x)= ---------------

x + 3
Answer: vertical asymptote is x=-3 because f(x) is not defined at that value.
If the degree of the numerator is bigger than the denominator, there is no horizontal asymptote.
for slant/oblique asymptote divide numerator by denominator
x+3 ) x^2-4x+5 (x-7
x^2+3x
___________________
-7x+5
-7x-21
__________________
26
y= f(x) = (x-7) + 26 / (x+3)

y= x-7 is the slant/oblique asymptote.

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