Solve: 2x-1 + 2x-4 + 2x-2 = 6,5555...
6,55555... = 6 + 5/9 = 54/9 + 5/9 = 59/9
The terms: 2x-1, 2x-4, and 2x-2 can all be written in terms of 2x-4:
2x-1 = 2x-4+3 = 2x-4·23 = 2x-4·8
2x-4 = 2x-4·1
2x-2 = 2x-4+2 = 2x-4·22 = 2x-4·4
Therefore, the problem can be re-written as:
2x-4·8 + 2x-4·1 + 2x-4·4 = 59/9
Factoring out the term 2x-4:
2x-4(8 + 1 + 4) = 59/9
2x-4(13) = 59/9
Dividing by 13:
2x-4 = (59/9) / 13
2x-4 = 59/117
Taking the log of both sides:
log( 2x-4 ) = log(59/117)
(x-4)·log(2) = log(59/117)
Dividing by log(2):
x - 4 = log(59/117) / log(2)
x = log(59/117) / log(2) + 4
x = 3.012278...
If the length of the base of an isosceles triangle is 'a' and the length of each of the two congruent sides is 'b', then the distance between the orthocenter (the point of intersection of the altitudes) and the circumcenter (the point of intersection of the perpendicular bisectors of the sides) can be found using the formula: | (b2 - a2) / sqrt( 4·b2 - a2 ) |.
I calculated this formula using analytic geometry.
Since Loc can mow the lawn in 20 minutes, his rate is 1 lawn / 20 minutes or 1/20th of the lawn per minute.
Since Reza can mow the lawn in 30 minutes, his rate is 1 lawn / 30 minutes or 1/30th of the lawn per minute.
Loc mows for 5 minutes before he his joined by Reza. In these 5 minutes, he can mow (1/20)·(5) = 5/20th = 1/4th of the lawn.
Now, there is only 3/4th of the lawn to finish.
Assuming that the time they mow together is x, we can create this equation:
Amount done by Loc + Amount done by Reza = Total Amount
(1/20)(x) + (1/30)(x) = 3/4
Multiplying both sides by 60:
60(1/20)(x) + 60(1/30)(x) = 60(3/4)
3x + 2x = 45
5x = 45
x = 9 [It will take them 9 minutes, working together, to finish the lawn.]
Adding 9 to the 5 minutes that Loc mows alone = 14 minutes after Loc starts ---> 1:14 pm
For the lines, x = a and x = -a, with a > 0: the graph will be a pair of vertical lines, one passing through the point (a,0) and the other passing through the point (-a,0). There will be no 'solution' beccause there is no point of intersection.
For the lines, y = b and y = -b, with b > 0: the graph will be a pair of horizontal lines, one passing through the point (0,b) and the other passing through the point (0, -b). There will be no 'solution' because there is no point of intersection.
A line segment starting at point A(2, -2) extends through point B(14, 4) to point C.
If the length from B to C is 1/3rd the length from A to B, what are the coordinates of point C?
The x-distance from A to B is 12 because 14 - 2 = 12.
The y-distance from A to B is 6 because 4 - -2 = 6.
Since 1/3rd of 12 is 4, the x-value of point C must be 4 more than the x-value of point B ---> 14 + 4 = 18.
Since 1/3rd of 6 is 2, the y-value of point C must be 2 more than the y-value of point B ---> 4 + 2 = 6.
Therefore, the coordinates of point C are (18, 6).
The probability of choosing a red marble from bag A: (1/3) x (35/100) = 7/60
The probability of choosing a red marble from bag B: (2/3) x (55/100) = 22/60
The probability of getting a red marble is 29/60.
Getting that red marble from bag A is 7/60 / 29/60 = 7/29.
Getting that red marble from bag B is 22/60 / 29/60 = 22/29.